As has been noted in the introductory section of this Recommendation, mathematics can be distinguished by its use of a (relatively) formal language, mathematical notation. However, mathematics and its presentation should not be viewed as one and the same thing. Mathematical sums or products exist and are meaningful to many applications completely without regard to how they are rendered aurally or visually. The intent of the content markup in the Mathematical Markup Language is to provide an explicit encoding of the underlying mathematical structure of an expression, rather than any particular rendering for the expression.
There are many reasons for providing a specific encoding for content. Even a disciplined and systematic use of presentation tags cannot properly capture this semantic information. This is because without additional information it is impossible to decide whether a particular presentation was chosen deliberately to encode the mathematical structure or simply to achieve a particular visual or aural effect. Furthermore, an author using the same encoding to deal with both the presentation and mathematical structure might find a particular presentation encoding unavailable simply because convention had reserved it for a different semantic meaning.
The difficulties stem from the fact that there are many to one mappings from presentation to semantics and vice versa. For example the mathematical construct " H multiplied by e" is often encoded using an explicit operator as in H × e. In different presentational contexts, the multiplication operator might be invisible " H e", or rendered as the spoken word "times". Generally, many different presentations are possible depending on the context and style preferences of the author or reader. Thus, given " H e" out of context it may be impossible to decide if this is the name of a chemical or a mathematical product of two variables H and e.
Mathematical presentation also changes with culture and time: some expressions in combinatorial mathematics today have one meaning to a Russian mathematician, and quite another to a French mathematician; see 5.4.1 Notational Style Sheets for an example. Notations may lose currency, for example the use of musical sharp and flat symbols to denote maxima and minima [Chaundy1954]. A notation in use in 1644 for the multiplication mentioned above was H e [Cajori1928].
When we encode the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually, we are able to interchange information more precisely with those systems that are able to manipulate the mathematics. In the trivial example above, such a system could substitute values for the variables H and e and evaluate the result. Further interesting application areas include interactive textbooks and other teaching aids.
The semantics of general mathematical notation is not a matter of consensus. It would be an enormous job to systematically codify most of mathematics  a task that can never be complete. Instead, MathML makes explicit a relatively small number of commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it provides a mechanism for associating semantics with new notational constructs. In this way, mathematical concepts that are not in the base collection of elements can still be encoded (4.2.6 Syntax and Semantics).
The base set of content elements is chosen to be adequate for simple coding of most of the formulas used from kindergarten to the end of high school in the United States, and probably beyond through the first two years of college, that is up to ALevel or Baccalaureate level in Europe. Subject areas covered to some extent in MathML are:
arithmetic, algebra, logic and relations
calculus and vector calculus
set theory
sequences and series
elementary classical functions
statistics
linear algebra
It is not claimed, or even suggested, that the proposed set of elements is complete for these areas, but the provision for author extensibility greatly alleviates any problem omissions from this finite list might cause.
The design of the MathML content elements are driven by the following principles:
The expression tree structure of a mathematical expression should be directly encoded by the MathML content elements.
The encoding of an expression tree should be explicit, and not dependent on the special parsing of PCDATA or on additional processing such as operator precedence parsing.
The basic set of mathematical content constructs that are provided should have default mathematical semantics.
There should be a mechanism for associating specific mathematical semantics with the constructs.
The primary goal of the content encoding is to establish explicit connections between mathematical structures and their mathematical meanings. The content elements correspond directly to parts of the underlying mathematical expression tree. Each structure has an associated default semantics and there is a mechanism for associating new mathematical definitions with new constructs.
Significant advantages to the introduction of contentspecific tags include:
Usage of presentation elements is less constrained. When mathematical semantics are inferred from presentation markup, processing agents must either be quite sophisticated, or they run the risk of inferring incomplete or incorrect semantics when irregular constructions are used to achieve a particular aural or visual effect.
It is immediately clear which kind of information is being encoded simply by the kind of elements that are used.
Combinations of semantic and presentation elements can be used to convey both the appearance and its mathematical meaning much more effectively than simply trying to infer one from the other.
Expressions described in terms of content elements must still be rendered. For common expressions, default visual presentations are usually clear. "Take care of the sense and the sounds will take care of themselves" wrote Lewis Carroll [Carroll1871]. Default presentations are included in the detailed description of each element occurring in 4.4 The Content Markup Elements.
To accomplish these goals, the MathML content encoding is based on the concept of an expression tree. A content expression tree is constructed from a collection of more primitive objects, referred to herein as containers and operators. MathML possesses a rich set of predefined container and operator objects, as well as constructs for combining containers and operators in mathematically meaningful ways. The syntax and usage of these content elements and constructions is described in the next section.
Since the intent of MathML content markup is to encode mathematical expressions in such a way that the mathematical structure of the expression is clear, the syntax and usage of content markup must be consistent enough to facilitate automated semantic interpretation. There must be no doubt when, for example, an actual sum, product or function application is intended and if specific numbers are present, there must be enough information present to reconstruct the correct number for purposes of computation. Of course, it is still up to a MathMLcompliant processor to decide what is to be done with such a contentbased expression, and computation is only one of many options. A renderer or a structured editor might simply use the data and its own builtin knowledge of mathematical structure to render the object. Alternatively, it might manipulate the object to build a new mathematical object. A more computationally oriented system might attempt to carry out the indicated operation or function evaluation.
The purpose of this section is to describe the intended, consistent usage. The requirements involve more than just satisfying the syntactic structure specified by an XML DTD. Failure to conform to the usage as described below will result in a MathML error, even though the expression may be syntactically valid according to the DTD.
In addition to the usage information contained in this section, 4.4 The Content Markup Elements gives a complete listing of each content element, providing reference information about their attributes, syntax, examples and suggested default semantics and renderings. The rules for using presentation markup within content markup are explained in 5.2.3 Presentation Markup Contained in Content Markup. An informal EBNF grammar describing the syntax for the content markup is given in B Content Markup Validation Grammar.
MathML content encoding is based on the concept of an expression tree. As a general rule, the terminal nodes in the tree represent basic mathematical objects, such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree generally represent some kind of function application or other mathematical construction that builds up a compound object. Function application provides the most important example; an internal node might represent the application of a function to several arguments, which are themselves represented by the terminal nodes underneath the internal node.
The MathML content elements can be grouped into the following categories based on their usage:
containers
operators and functions
qualifiers
relations
conditions
semantic mappings
constants and symbols
These are the building blocks out of which MathML content expressions are constructed. Each category is discussed in a separate section below. In the remainder of this section, we will briefly introduce some of the most common elements of each type, and consider the general constructions for combining them in mathematically meaningful ways.
Content expression trees are built up from basic mathematical objects. At the lowest level,
leaf nodes are encapsulated in nonempty elements that define their type. Numbers and symbols are marked by the
token elements
cn
and
ci
. More elaborate constructs such as sets, vectors and matrices are also marked using elements to denote their types, but rather
than containing data directly, these
container elements are constructed out of other elements. Elements are used in order to clearly identify the underlying objects. In
this way, standard XML parsing can be used and attributes can be used to specify global properties of the objects.
The containers such as
<cn>12345</cn>
,
<ci>x</ci>
and
<csymbol definitionURL="mySymbol.htm" encoding="text">S</csymbol>
represent mathematical numbers , identifiers and externally defined symbols. Below, we will look at
operator
elements such as
plus
or
sin
, which provide access to the basic mathematical operations and functions applicable to those objects. Additional containers
such as
set
for sets, and
matrix
for matrices are provided for representing a variety of common compound objects.
For example, the number 12345 is encoded as
<cn>12345</cn>
The attributes and
PCDATA content together provide the data necessary for an application to parse the number. For example, a default base of 10 is
assumed, but to communicate that the underlying data was actually written in base 8, simply set the
base
attribute to 8 as in
<cn base="8">12345</cn>
while the complex number 3 + 4i can be encoded as
<cn type="complexcartesian">3<sep/>4</cn>
Such information makes it possible for another application to easily parse this into the correct number.
As another example, the scalar symbol v is encoded as
<ci>v</ci>
By default,
ci
elements represent elements from a commutative field (see
C Content Element Definitions). If a vector is intended then this fact can be encoded as
<ci type="vector">v</ci>
This invokes default semantics associated with the
vector
element, namely an arbitrary element of a finitedimensional vector space.
By using the
ci
and
csymbol
elements we have made clear that we are referring to a mathematical identifier or symbol but this does not say anything about
how it should be rendered. By default a symbol is rendered as if the
ci
or
csymbol
element were actually the presentation element
mi
(see
3.2.3 Identifier (mi)). The actual rendering of a mathematical symbol can be made as elaborate as necessary simply by using the more elaborate
presentational constructs (as described in
3 Presentation Markup) in the body of the
ci
or
csymbol
element.
The default rendering of a simple
cn
tagged object is the same as for the presentation element
mn
with some provision for overriding the presentation of the
PCDATA by providing explicit
mn
tags. This is described in detail in
4.4 The Content Markup Elements.
The issues for compound objects such as sets, vectors and matrices are all similar to those outlined above for numbers and symbols. Each such object has global properties as a mathematical object that impact how it is to be parsed. This may affect everything from the interpretation of operations that are applied to it to how to render the symbols representing it. These mathematical properties are captured by setting attribute values.
The notion of constructing a general expression tree is essentially that of applying an operator to subobjects. For example,
the sum
a +
b can be thought of as an application of the addition operator to two arguments
a and
b. In MathML, elements are used for operators for much the same reason that elements are used to contain objects. They are
recognized at the level of XML parsing, and their attributes can be used to record or modify the intended semantics. For example,
with the MathML
plus
element, setting the
definitionURL
and
encoding
attributes as in
<plus definitionURL="www.example.com/VectorCalculus.htm" encoding="text"/>
can communicate that the intended operation is vectorbased.
There is also another reason for using elements to denote operators. There is a crucial semantic distinction between the function
itself and the expression resulting from applying that function to zero or more arguments which must be captured. This is
addressed by making the functions selfcontained objects with their own properties and providing an explicit
apply
construct corresponding to function application. We will consider the
apply
construct in the next section.
MathML contains many predefined operator elements, covering a range of mathematical subjects. However, an important class
of expressions involve unknown or userdefined functions and symbols. For these situations, MathML provides a general
csymbol
element, which is discussed below.
apply
construct
The most fundamental way of building up a mathematical expression in MathML content markup is the
apply
construct. An
apply
element typically applies an operator to its arguments. It corresponds to a complete mathematical expression. Roughly speaking,
this means a piece of mathematics that could be surrounded by parentheses or
"logical brackets" without changing its meaning.
For example, (x + y) might be encoded as
<apply> <plus/> <ci> x </ci> <ci> y </ci> </apply>
The opening and closing tags of
apply
specify exactly the scope of any operator or function. The most typical way of using
apply
is simple and recursive. Symbolically, the content model can be described as:
<apply> op a b </apply>
where the
operands a and b are containers or other contentbased elements themselves, and
op is an operator or function. Note that since
apply
is a container, this allows
apply
constructs to be nested to arbitrary depth.
An
apply
may in principle have any number of operands:
<apply> op a b [c...] <apply>
For example, (x + y + z) can be encoded as
<apply> <plus/> <ci> x </ci> <ci> y </ci> <ci> z </ci> </apply>
Mathematical expressions involving a mixture of operations result in nested occurrences of
apply
. For example,
a
x +
b would be encoded as
<apply> <plus/> <apply> <times/> <ci> a </ci> <ci> x </ci> </apply> <ci> b </ci> </apply>
There is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly.
The
apply
tags provide the proper grouping for the reuse of the expressions within other constructs. Any expression enclosed by an
apply
element is viewed as a single coherent object.
An expression such as (F + G)(x) might be a product, as in
<apply> <times/> <apply> <plus/> <ci> F </ci> <ci> G </ci> </apply> <ci> x </ci> </apply>
or it might indicate the application of the function F + G to the argument x. This is indicated by constructing the sum
<apply> <plus/> <ci> F </ci> <ci> G </ci> </apply>
and applying it to the argument x as in
<apply> <apply> <plus/> <ci> F </ci> <ci> G </ci> </apply> <ci> x </ci> </apply>
Both the function and the arguments may be simple identifiers or more complicated expressions.
In MathML 1.0 , another construction closely related to the use of the
apply
element with operators and arguments was the
reln
element. The
reln
element was used to denote that a mathematical relation holds between its arguments, as opposed to applying an operator.
Thus, the MathML markup for the expression
x <
y was given in MathML 1.0 by:
<reln> <lt/> <ci> x </ci> <ci> y </ci> </reln>
In MathML 2.0, the
apply
construct is used with all operators, including logical operators. The expression above becomes
<apply> <lt/> <ci> x </ci> <ci> y </ci> </apply>
in MathML 2.0. The use of
reln
with relational operators is supported
for reasons of backwards compatibility, but deprecated. Authors creating new content are
encouraged to use
apply
in all cases.
The most common operations and functions such as
plus
and
sin
have been predefined explicitly as empty elements (see
4.4 The Content Markup Elements). They have
type
and
definitionURL
attributes, and by changing these attributes, the author can record that a different sort of algebraic operation is intended.
This allows essentially the same notation to be reused for a discussion taking place in a different algebraic domain.
Due to the nature of mathematics the notation must be extensible. The key to extensibility is the ability of the user to define new functions and other symbols to expand the terrain of mathematical discourse.
It is always possible to create arbitrary expressions, and then to use them as symbols in the language. Their properties can
then be inferred directly from that usage as was done in the previous section. However, such an approach would preclude being
able to encode the fact that the construct was a known symbol, or to record its mathematical properties except by actually
using it. The
csymbol
element is used as a container to construct a new symbol in much the same way that
ci
is used to construct an identifier. (Note that
"symbol" is used here in the abstract sense and has no connection with any presentation of the construct on screen or paper).
The difference in usage is that
csymbol
should refer to some mathematically defined concept with an external definition referenced via the
definitionURL
attribute, whereas
ci
is used for identifiers that are essentially
"local" to the MathML expression and do not use any external definition mechanism. The target of the
definitionURL
attribute on the
csymbol
element may encode the definition in any format; the particular encoding in use is given by the
encoding
attribute.
To use
csymbol
to describe a completely new function, we write for example
<csymbol definitionURL="www.example.com/VectorCalculus.htm" encoding="text"> Christoffel </csymbol>
The
definitionURL
attribute specifies a URI that provides a written definition for the
Christoffel symbol. Suggested default definitions for the content elements of MathML appear in
C Content Element Definitions in a format based on OpenMath, although there is no requirement that a particular format be used. The role of the
definitionURL
attribute is very similar to the role of definitions included at the beginning of many mathematical papers, and which often
just refer to a definition used by a particular book.
MathML 1.0 supported the use of the fn to encode the fact that a construct is explicitly being used as a function or operator. To record the fact that F+ G is being used semantically as if it were a function, it was encoded as:
<fn> <apply> <plus/> <ci>F</ci> <ci>G</ci> </apply> </fn>
This usage, although allowed in MathML 2.0 for reasons of backwards compatibility,
is now deprecated.
The fact that a construct is being used as an operator is clear from the position of the construct as the
first child of the
apply
. If it is required to add additional information to the construct, it should be wrapped in a
semantics
element, for example:
<semantics definitionURL="www.example.com/vectorfuncs/plus.htm" encoding="Mathematica"> <apply> <plus/> <ci>F</ci> <ci>G</ci> </apply> </semantics>
MathML 1.0 supported the use of
definitionURL
with
fn to refer to external definitions for userdefined
functions. This usage, although allowed for reasons of backwards
compatibility, is deprecated in
MathML 2.0 in favor of using
csymbol
to define the function, and then
apply
to link the function to its arguments. For example:
<apply> <csymbol definitionURL="http://www.example.org/function_spaces.html#my_def" encoding="text"> BigK </csymbol> <ci>x</ci> <ci>y</ci> </apply>
Given functions, it is natural to have functional inverses. This is handled by the
inverse
element.
Functional inverses can be problematic from a mathematical point of view in that they implicitly involve the definition of an inverse for an arbitrary function F. Even at the Kthrough12 level the concept of an inverse F ^{1} of many common functions F is not used in a uniform way. For example, the definitions used for the inverse trigonometric functions may differ slightly depending on the choice of domain and/or branch cuts.
MathML adopts the view: if F is a function from a domain D to D', then the inverse G of F is a function over D' such that G(F(x)) = x for x in D. This definition does not assert that such an inverse exists for all or indeed any x in D, or that it is singlevalued anywhere. Also, depending on the functions involved, additional properties such as F(G(y)) = y for y in D' may hold.
The
inverse
element is applied to a function whenever an inverse is required. For example, application of the inverse sine function to
x, i.e. sin^{1} (x), is encoded as:
<apply> <apply> <inverse/> <sin/> </apply> <ci> x </ci> </apply>
While
arcsin
is one of the predefined MathML functions, an explicit reference to sin^{1}(x) might occur in a document discussing possible definitions of
arcsin
.
Consider a document discussing the vectors
A = (a, b,
c) and
B = (d, e,
f), and later including the expression
V =
A +
B. It is important to be able to communicate the fact that wherever
A and
B are used they represent a particular vector. The properties of that vector may determine aspects of operators such as
plus
.
The simple fact that A is a vector can be communicated by using the markup
<ci type="vector">A</ci>
but this still does not communicate, for example, which vector is involved or its dimensions.
The declare
construct is used to associate
specific properties or meanings with an object. The actual declaration
itself is not rendered visually (or in any other form). However, it
indirectly impacts the semantics of all affected uses of the declared
object.
Declarations must occur at the beginning of a math
element. The scope of a declaration is the entire
math
element in which the declaration is made.
The scope
attribute of a declare
may be included but has no effect since the two possible values of
"local" or "global"
now have the same meaning. The "global" attribute value
is still allowed for backwards compatibility with MathML 1.0.,
but is deprecated in MathML 2.0.
The uses of the declare
element range from
resetting default attribute values to associating an expression with a
particular instance of a more elaborate structure. Subsequent uses of the
original expression (within the scope of the declare
) play the same semantic role as would the
paired object.
For example, the declaration
<declare> <ci> A </ci> <vector> <ci> a </ci> <ci> b </ci> <ci> c </ci> </vector> </declare>
specifies that A stands for the particular vector (a,
b, c) so that subsequent uses of A as in
V = A + B can take this into account. When declare
is used in this way, the actual encoding
<apply> <eq/> <ci> V </ci> <apply> <plus/> <ci> A </ci> <ci> B </ci> </apply> </apply>
remains unchanged but the expression can be interpreted properly as vector addition.
There is no requirement to declare an expression to stand for a specific object. For example, the declaration
<declare type="vector"> <ci> A </ci> </declare>
specifies that
A is a vector without indicating the number of components or the values of specific components. The possible values for the
type
attribute include all the predefined container element names such as
vector
,
matrix
or
set
(see
4.3.2.9
type).
The lambda calculus allows a user to construct a function from a variable and an expression. For example, the lambda construct underlies the common mathematical idiom illustrated here:
Let f be the function taking x to x^{2} + 2
There are various notations for this concept in mathematical literature, such as (x, F(x)) = F or (x, [F]) =F, where x is a free variable in F.
This concept is implemented in MathML with the lambda
element. A lambda construct with n
internal variables is encoded by a lambda
element
with n+1 children. All but the last child must be bvar
elements containing the identifiers of the
internal variables. The last child is an expression defining the
function. This is typically an apply
, but can also
be any container element.
The following constructs (x, sin(x+1)):
<lambda> <bvar><ci> x </ci></bvar> <apply> <sin/> <apply> <plus/> <ci> x </ci> <cn> 1 </cn> </apply> </apply> </lambda>
To use
declare
and
lambda
to construct the function
f for which
f(
x) =
x^{2} +
x + 3 use:
<declare type="fn"> <ci> f </ci> <lambda> <bvar><ci> x </ci></bvar> <apply> <plus/> <apply> <power/> <ci> x </ci> <cn> 2 </cn> </apply> <ci> x </ci> <cn> 3 </cn> </apply> </lambda> </declare>
The following markup declares and constructs the function J such that J(x, y) is the integral from x to y of t^{4} with respect to t.
<declare type="fn"> <ci> J </ci> <lambda> <bvar><ci> x </ci></bvar> <bvar><ci> y </ci></bvar> <apply> <int/> <bvar> <ci> t </ci> </bvar> <lowlimit> <ci> x </ci> </lowlimit> <uplimit> <ci> y </ci> </uplimit> <apply> <power/> <ci>t</ci> <cn>4</cn> </apply> </apply> </lambda> </declare>
The function J can then in turn be applied to an argument pair.
The last example of the preceding section illustrates the use of
qualifier elements
lowlimit
,
uplimit
, and
bvar
used in conjunction with the
int
element. A number of common mathematical constructions involve additional data that is either implicit in conventional notation,
such as a bound variable, or thought of as part of the operator rather than an argument, as is the case with the limits of
a definite integral.
Content markup uses qualifier elements in conjunction with a number of operators, including integrals, sums, series, and certain
differential operators. Qualifier elements appear in the same
apply
element with one of these operators. In general, they must appear in a certain order, and their precise meaning depends on
the operators being used. For details, see
4.2.3.2 Operators taking Qualifiers.
The qualifier element
bvar
is also used in another important MathML construction. The
condition
element is used to place conditions on bound variables in other expressions. This allows MathML to define sets by rule, rather
than enumeration, for example. The following markup, for instance, encodes the set {
x 
x < 1}:
<set> <bvar><ci> x </ci></bvar> <condition> <apply> <lt/> <ci> x </ci> <cn> 1 </cn> </apply> </condition> </set>
While the primary role of the MathML content element set is to directly encode the mathematical structure of expressions independent of the notation used to present the objects, rendering issues cannot be ignored. Each content element has a default rendering, given in 4.4 The Content Markup Elements, and several mechanisms (including 4.3.3.2 General Attributes) are provided for associating a particular rendering with an object.
Containers provide a means for the construction of mathematical objects of a given type.
Tokens 
ci ,
cn ,
csymbol 
Constructors 
interval ,
list ,
matrix ,
matrixrow ,
set ,
vector ,
apply ,
reln ,
fn ,
lambda ,
piecewise, piece, otherwise

Specials 
declare 
Token elements are typically the leaves of the MathML expression tree. Token elements are used to indicate mathematical identifiers, numbers and symbols.
It is also possible for the canonically empty operator elements such as
exp
,
sin
and
cos
to be leaves in an expression tree. The usage of operator elements is described in
4.2.3 Functions, Operators and Qualifiers.
The
cn
element is the MathML token element used to represent numbers. The supported types of numbers include:
"real",
"integer",
"rational",
"complexcartesian", and
"complexpolar", with
"real" being the default type. An attribute
base
(with default value
"10") is used to help specify how the content is to be parsed. The content itself is essentially
PCDATA, separated by
sep
when two parts are needed in order to fully describe a number. For example, the real number 3 is constructed by
<cn type="real"> 3 </cn>
, while the rational number 3/4 is constructed as
<cn type="rational"> 3<sep/>4 </cn>
. The detailed structure and specifications are provided in
4.4.1.1 Number (cn).
The
ci
element, or
"content identifier" is used to construct a variable, or an identifier. A
type
attribute indicates the type of object the symbol represents. Typically,
ci
represents a real scalar, but no default is specified. The content is either
PCDATA or a general presentation construct (see
3.1.6 Summary of Presentation Elements). For example,
<ci> <msub> <mi>c</mi> <mn>1</mn> </msub> </ci>
encodes an atomic symbol that displays visually as c_{1} which, for purposes of content, is treated as a single symbol representing a real number. The detailed structure and specifications are provided in 4.4.1.2 Identifier (ci).
The
csymbol
element, or
"content symbol" is used to construct a symbol whose semantics are not part of the core content elements provided by MathML,
but defined externally.
csymbol
does not make any attempt to describe how to map the arguments occurring in any application of the function into a new MathML
expression. Instead, it depends on its
definitionURL
attribute to point to a particular meaning, and the
encoding
attribute to give the syntax of this definition. The content of a
csymbol
is either
PCDATA or a general presentation construct (see
3.1.6 Summary of Presentation Elements). For example,
<csymbol definitionURL="www.example.com/ContDiffFuncs.htm" encoding="text"> <msup> <mi>C</mi> <mn>2</mn> </msup> </csymbol>
encodes an atomic symbol that displays visually as C^{2} and that, for purposes of content, is treated as a single symbol representing the space of twicedifferentiable continuous functions. The detailed structure and specifications are provided in 4.4.1.3 Externally defined symbol (csymbol).
MathML provides a number of elements for combining elements into familiar compound objects. The compound objects include things like lists and sets. Each constructor produces a new type of object.
The
interval
element is described in detail in
4.4.2.4 Interval (interval). It denotes an interval on the real line with the values represented by its children as end points. The
closure
attribute is used to qualify the type of interval being represented. For example,
<interval closure="openclosed"> <ci> a </ci> <ci> b </ci> </interval>
represents the openclosed interval often written (a, b].
The
set
and
list
elements are described in detail in
4.4.6.1 Set (set) and
4.4.6.2 List (list). Typically, the child elements of a possibly empty
list
element are the actual components of an ordered
list. For example, an ordered list of the three symbols
a,
b, and
c is encoded as
<list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list>
Alternatively,
bvar
and
condition
elements can be used to define lists where membership depends on satisfying certain conditions.
An
order
attribute can be used to specify what ordering is to be used. When the nature of the child elements permits, the ordering
defaults to a numeric or lexicographic ordering.
Sets are structured much the same as lists except that there is no implied ordering and the
type
of set may be
"normal" or
"multiset" with
"multiset" indicating that repetitions are allowed.
For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted. For example, one might construct a list of equations, or of inequalities.
The
matrix
element is used to represent mathematical matrices. It is described in detail in
4.4.10.2 Matrix (matrix). It has zero or more child elements, all of which are
matrixrow
elements. These in turn expect zero or more child elements that evaluate to algebraic expressions or numbers. These subelements
are often real numbers, or symbols as in
<matrix> <matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow> <matrixrow> <cn> 3 </cn> <cn> 4 </cn> </matrixrow> </matrix>
The
matrixrow
elements must always be contained inside of a matrix, and all rows in a given matrix must have the same number of elements.
Note that the behavior of the
matrix
and
matrixrow
elements is substantially different from the
mtable
and
mtr
presentation elements.
The
vector
element is described in detail in
4.4.10.1 Vector (vector). It constructs vectors from an
ndimensional vector space so that its
n child elements typically represent real or complex valued scalars as in the threeelement vector
<vector> <apply> <plus/> <ci> x </ci> <ci> y </ci> </apply> <cn> 3 </cn> <cn> 7 </cn> </vector>
The
apply
element is described in detail in
4.4.2.1 Apply (apply). Its purpose is to apply a function or operator to its arguments to produce an expression representing an element of the
codomain of the function. It is involved in everything from forming sums such as
a +
b as in
<apply> <plus/> <ci> a </ci> <ci> b </ci> </apply>
through to using the sine function to construct sin(a) as in
<apply> <sin/> <ci> a </ci> </apply>
or constructing integrals. Its usage in any particular setting is determined largely by the properties of the function (the first child element) and as such its detailed usage is covered together with the functions and operators in 4.2.3 Functions, Operators and Qualifiers.
The
reln
element is described in detail in
4.4.2.2 Relation (reln). It was used in MathML 1.0 to construct an expression such as
a =
b, as in
<reln><eq/> <ci> a </ci> <ci> b </ci> </reln>
indicating an intended comparison between two mathematical values.
MathML 2.0 takes the view that this should be regarded as the application of a boolean function, and as such could be constructed
using
apply
. The use of
reln
with logical operators is supported
for reasons of backwards compatibility, but deprecated in favor of
apply
.
The
fn
element was used in MathML 1.0 to make
explicit the fact that an expression is being used as a function or
operator. This is allowed in MathML 2.0 for backwards compatibility,
but is deprecated, as the use of
an expression as a function or operator is clear from its position as
the first child of an
apply
.
fn
is discussed in detail in
4.4.2.3 Function (fn).
The
lambda
element is used to construct a userdefined function from an expression and one or more free variables. The lambda construct
with
n internal variables takes
n+1 children. The first (second, up to
n) is a
bvar
containing the identifiers of the internal variables. The last is an expression defining the function. This is typically
an
apply
, but can also be any container element. The following constructs
(x, sin
x)
<lambda> <bvar><ci> x </ci></bvar> <apply> <sin/> <ci> x </ci> </apply> </lambda>
The following constructs the constant function (x, 3)
<lambda> <bvar><ci> x </ci></bvar> <cn> 3 </cn> </lambda>
The
piecewise
,
piece
,
otherwise
elements are used to support "piecewise" declarations of the form "
H(x) = 0 if x less than 0,
H(x) = 1 otherwise".
<piecewise> <piece> <cn> 0 </cn> <apply><lt/><ci> x </ci> <cn> 0 </cn></apply> </piece> <otherwise> <ci> x </ci> </otherwise> </piecewise>
The piecewise
elements are discussed in detail in
4.4.2.16 Piecewise declaration
(piecewise, piece,
otherwise)
.
The
declare
construct is described in detail in
4.4.2.8 Declare (declare). It is special in that its entire purpose is to modify
the semantics of other objects. It is not rendered visually or aurally.
The need for declarations arises any time a symbol (including more general presentations) is being used to represent an instance of an object of a particular type. For example, you may wish to declare that the symbolic identifier V represents a vector. The single argument form can be used to set properties of objects by setting the default values of implied attribute values to specific values.
The declaration
<declare type="vector"><ci>V</ci></declare>
resets the default type attribute of
<ci>V</ci>
to
"vector" for all affected occurrences of
<ci>V</ci>
. This avoids having to write
<ci type="vector">V</ci>
every time you use the symbol.
More generally,
declare
can be used to associate expressions with specific content. For example, the declaration
<declare> <ci>F</ci> <lambda> <bvar><ci> U </ci></bvar> <apply> <int/> <bvar><ci> x </ci></bvar> <lowlimit><cn> 0 </cn></lowlimit> <uplimit><ci> a </ci></uplimit> <ci> U </ci> </apply> </lambda> </declare>
associates the symbol
F with a new function defined by the
lambda
construct. Within the scope where the declaration is in effect, the expression
<apply> <ci>F</ci> <ci> U </ci> </apply>
stands for the integral of U from 0 to a.
The
declare
element can also be used to change the definition of a function or operator. For example, if the URL
http://.../MathML:noncommutplus
described a noncommutative plus operation encoded in Maple syntax, then the declaration
<declare definitionURL="http://.../MathML:noncommutplus" encoding="Maple"> <plus/> </declare>
would indicate that all affected uses of
plus
are to be interpreted as having that definition of
plus
.
The operators and functions defined by MathML can be divided into categories as shown in the table below.
unary arithmetic 
factorial ,
minus ,
abs ,
conjugate ,
arg ,
real ,
imaginary ,
floor ,
ceiling

unary logical 
not 
unary functional 
inverse ,
ident ,
domain ,
codomain ,
image

unary elementary classical functions 
sin ,
cos ,
tan ,
sec ,
csc ,
cot ,
sinh ,
cosh ,
tanh ,
sech ,
csch ,
coth ,
arcsin ,
arccos ,
arctan ,
arccosh ,
arccot ,
arccoth ,
arccsc ,
arccsch ,
arcsec ,
arcsech ,
arcsinh ,
arctanh ,
exp ,
ln ,
log 
unary linear algebra 
determinant ,
transpose 
unary calculus and vector calculus 
divergence ,
grad ,
curl ,
laplacian 
unary settheoretic 
card 
binary arithmetic 
quotient ,
divide ,
minus ,
power ,
rem 
binary logical 
implies ,
equivalent ,
approx 
binary set operators 
setdiff 
binary linear algebra 
vectorproduct ,
scalarproduct ,
outerproduct 
nary arithmetic 
plus ,
times ,
max ,
min ,
gcd ,
lcm

nary statistical 
mean ,
sdev ,
variance ,
median ,
mode 
nary logical 
and ,
or ,
xor 
nary linear algebra 
selector 
nary set operator 
union ,
intersect ,
cartesianproduct

nary functional 
fn ,
compose 
integral, sum, product operators 
int ,
sum ,
product 
differential operator 
diff ,
partialdiff 
quantifier 
forall ,
exists 
From the point of view of usage, MathML regards functions (for example
sin
and
cos
) and operators (for example
plus
and
times
) in the same way. MathML predefined functions and operators are all canonically empty elements.
Note that the
csymbol
element can be used to construct a userdefined symbol that can be used as a function or operator.
MathML functions can be used in two ways. They can be used as the operator within an
apply
element, in which case they refer to a function evaluated at a specific value. For example,
<apply> <sin/> <cn>5</cn> </apply>
denotes a real number, namely sin(5).
MathML functions can also be used as arguments to other operators, for example
<apply> <plus/><sin/><cos/> </apply>
denotes a function, namely the result of adding the sine and cosine functions in some function space. (The default semantic
definition of
plus
is such that it infers what kind of operation is intended from the type of its arguments.)
The number of child elements in the
apply
is defined by the element in the first (i.e. operator) position.
Unary operators are followed by exactly one other child element within the
apply
.
Binary operators are followed by exactly two child elements.
Nary operators are followed by two or more child elements.
The one exception to these rules is that
declare
elements may be inserted in any position except the first.
declare
elements are not counted when satisfying the child element count for an
apply
containing a unary or binary operator element.
Integral, sum, product and differential operators are discussed below in 4.2.3.2 Operators taking Qualifiers.
The table below contains the qualifiers and the operators defined as taking qualifiers in MathML.
qualifiers 
lowlimit ,
uplimit ,
bvar ,
degree ,
logbase ,
interval ,
condition ,
domainofapplication ,
momentabout

operators 
int ,
sum ,
product ,
root ,
diff ,
partialdiff ,
limit ,
log ,
moment ,
min ,
max ,
forall ,
exists 
Operators taking qualifiers are canonically empty functions that differ from ordinary empty functions only in that they support
the use of special
qualifier elements to specify their meaning more fully. They are used in exactly the same way as ordinary operators, except that when
they are used as operators, certain qualifier elements are also permitted to be in the enclosing
apply
. Qualifiers always follow the operator and precede the argument if it is present. If more than one qualifier is present,
they appear in the order
bvar
,
lowlimit
,
uplimit
,
interval
,
condition
,
domainofapplication
,
degree
,
momentabout
,
logbase
. A typical example is:
<apply> <int/> <bvar><ci>x</ci></bvar> <interval><cn>0</cn><cn>1</cn></interval> <apply> <power/> <ci>x</ci> <cn>2</cn> </apply> </apply>
It is also valid to use qualifier schema with a function not applied to an argument. For example, a function acting on integrable functions on the interval [0,1] might be denoted:
<fn> <apply> <int/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>1</cn></uplimit> </apply> </fn>
In addition to the defined usage in MathML, qualifier schema may be used with any userdefined symbol
(e.g. using csymbol
) or construct. The meaning of such a usage is not defined by MathML;
it would normally be userdefined using the definitionURL
attribute.
The meaning and usage of qualifier schema varies from function to function. The following list summarizes the usage of qualifier schema with the MathML functions taking qualifiers.
The
int
function accepts the
lowlimit
,
uplimit
,
bvar
,
interval
,
condition
and
domainofapplication
schemata. If both
lowlimit
and
uplimit
schemata are present, they denote the limits of a definite integral. The domain of integration may alternatively be specified
using
interval
,
condition
or
domainofapplication
. The
bvar
schema signifies the variable of integration. When used with
int
, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.
The
diff
function accepts the
bvar
schema. The
bvar
schema specifies with respect to which variable the derivative is being taken. The
bvar
may itself contain a
degree
schema that is used to specify the order of the derivative, i.e. a first derivative, a second derivative, etc. For example,
the second derivative of
f with respect to
x is:
<apply> <diff/> <bvar> <ci> x </ci> <degree> <cn> 2 </cn> </degree> </bvar> <apply><fn><ci>f</ci></fn> <ci> x </ci> </apply> </apply>
The
partialdiff
operator accepts zero or more
bvar
schemata, and an optional degree
qualifier schema. The
bvar
schema specify, in order, the variables with respect to which the derivative is being taken. Each
bvar
element may contain a
degree
schema which is used to specify the order of the derivative being taken with respect to that
variable. The optional degree
schema qualifier associated with the
partialdiff
element itself (that is, appearing as a child of the enclosing
apply
element rather than of one of the bvar
qualifiers) is used to represent
the total degree of the differentiation. Each
degree
schema used with partialdiff
is expected
to contain a single child schema. For example,
<apply> <partialdiff/> <bvar> <degree><cn>2</cn></degree> <ci>x</ci> </bvar> <bvar><ci>y</ci></bvar> <bvar><ci>x</ci></bvar> <degree><cn>4</cn></degree> <ci type="fn">f</ci> </apply>
denotes the mixed partial derivative ( d^{4} / d^{2}x dy dx ) f.
The
sum
and
product
functions accept the
bvar
,
lowlimit
,
uplimit
,
interval
,
condition
and
domainofapplication
schemata. If both
lowlimit
and
uplimit
schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the
interval
,
condition
or
domainofapplication
schema. The
bvar
schema signifies the internal variable in the sum or product. A typical example might be:
<apply> <sum/> <bvar><ci>i</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>100</cn></uplimit> <apply> <power/> <ci>x</ci> <ci>i</ci> </apply> </apply>
When used with
sum
or
product
, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.
The
limit
function accepts zero or more
bvar
schemata, and optional
condition
and
lowlimit
schemata. A
condition
may be used to place constraints on the
bvar
. The
bvar
schema denotes the variable with respect to which the limit is being taken. The
lowlimit
schema denotes the limit point. When used with
limit
, the
bvar
and
lowlimit
schemata are expected to contain a single child schema; otherwise an error is generated.
The
log
function accepts only the
logbase
schema. If present, the
logbase
schema denotes the base with respect to which the logarithm is being taken. Otherwise, the log is assumed to be base 10.
When used with
log
, the
logbase
schema is expected to contain a single child schema; otherwise an error is generated.
The
moment
function accepts the
degree
and momentabout
schema. If present, the
degree
schema denotes the order of the moment. Otherwise, the moment is assumed to be the first order moment. When used with
moment
, the
degree
schema is expected to contain a single child schema; otherwise an error is generated. If present, the
momentabout
schema denotes the point about which the moment is taken. Otherwise, the moment is assumed to be the moment about zero.
The
min
and
max
functions accept a
bvar
schema in cases where the maximum or minimum is being taken over a set of values specified by a
condition
schema together with an expression to be evaluated on that set.
In MathML1.0, the
bvar
element was optional when using a
condition
; if a
condition
element containing a single variable was given by itself following a
min
or
max
operator, the variable was implicitly
assumed to be bound, and the expression to be maximized or minimized
(if absent) was assumed to be the single bound variable. This usage
is deprecated in MathML 2.0 in
favor of explicitly stating the bound variable(s) and the expression
to be maximized or minimized in all cases.
The
min
and
max
elements may also be applied to a list of values in which case no qualifier schemata are used. For examples of all three
usages, see
4.4.3.4 Maximum and minimum (max,
min).
The universal and existential quantifier operators
forall
and
exists
are used in conjunction with one or more
bvar
schemata to represent simple logical assertions. There are two ways of using the logical quantifier operators. The first
usage is for representing a simple, quantified assertion. For example, the statement
"there exists
x< 9" would be represented as:
<apply> <exists/> <bvar><ci> x </ci></bvar> <apply><lt/> <ci> x </ci><cn> 9 </cn> </apply> </apply>
The second usage is for representing implications. Hypotheses are given by a
condition
element following the bound variables. For example the statement
"for all
x < 9,
x < 10" would be represented as:
<apply> <forall/> <bvar><ci> x </ci></bvar> <condition> <apply><lt/> <ci> x </ci><cn> 9 </cn> </apply> </condition> <apply><lt/> <ci> x </ci><cn> 10 </cn> </apply> </apply>
Note that in both usages one or more
bvar
qualifiers are mandatory.
binary relation 
neq ,
equivalent ,
approx ,
factorof

binary logical relation 
implies 
binary set relation 
in ,
notin ,
notsubset ,
notprsubset 
binary series relation 
tendsto 
nary relation 
eq ,
leq ,
lt ,
geq ,
gt 
nary set relation 
subset ,
prsubset 
The MathML content tags include a number of canonically empty elements which denote arithmetic and logical relations. Relations are characterized by the fact that, if an external application were to evaluate them (MathML does not specify how to evaluate expressions), they would typically return a truth value. By contrast, operators generally return a value of the same type as the operands. For example, the result of evaluating a < b is either true or false (by contrast, 1 + 2 is again a number).
Relations are bracketed with their arguments using the
apply
element in the same way as other functions. In MathML 1.0, relational operators were bracketed using
reln
. This usage, although still supported,
is now deprecated in favor of
apply
. The element for the relational operator is the first child element of the
apply
. Thus, the example from the preceding paragraph is properly marked up as:
<apply> <lt/> <ci>a</ci> <ci>b</ci> </apply>
It is an error to enclose a relation in an element other than
apply
or
reln
.
The number of child elements in the
apply
is defined by the element in the first (i.e. relation) position.
Unary relations are followed by exactly one other child element within the
apply
.
Binary relations are followed by exactly two child elements.
Nary relations are followed by zero or more child elements.
The one exception to these rules is that
declare
elements may be inserted in any position except the first.
declare
elements are not counted when satisfying the child element count for an
apply
containing a unary or binary relation element.
condition 
condition 
The
condition
element is used to define the
"such that" construct in mathematical expressions. Condition elements are used in a number of contexts in MathML. They are
used to construct objects like sets and lists by rule instead of by enumeration. They can be used with the
forall
and
exists
operators to form logical expressions. And finally, they can be used in various ways in conjunction with certain operators.
For example, they can be used with an
int
element to specify domains of integration, or to specify argument lists for operators like
min
and
max
.
The
condition
element is always used together with one or more
bvar
elements.
The exact interpretation depends on the context, but generally speaking, the
condition
element is used to restrict the permissible values of a bound variable appearing in another expression to those that satisfy
the relations contained in the
condition
. Similarly, when the
condition
element contains a
set
, the values of the bound variables are restricted to that set.
A condition element contains a single child that is either an
apply
, or a
reln
element (deprecated). Compound conditions are
indicated by applying relations such as
and
inside the child of the condition.
The following encodes "there exists x such that x^{5} < 3".
<apply> <exists/> <bvar><ci> x </ci></bvar> <condition> <apply><lt/> <apply> <power/> <ci>x</ci> <cn>5</cn> </apply> <cn>3</cn> </apply> </condition> </apply>
The next example encodes "for all x in N there exist prime numbers p, q such that p+q = 2x".
<apply> <forall/> <bvar><ci>x</ci></bvar> <condition> <apply><in/> <ci>x</ci> <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/setname1.ocd"> N </csymbol> </apply> </condition> <apply><exists/> <bvar><ci>p</ci></bvar> <bvar><ci>q</ci></bvar> <condition> <apply><and/> <apply><in/><ci>p</ci> <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/setname1.ocd"> P </csymbol> </apply> <apply><in/><ci>q</ci> <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/setname1.ocd"> P </csymbol> </apply> <apply><eq/> <apply><plus/><ci>p</ci><ci>q</ci></apply> <apply><times/><cn>2</cn><ci>x</ci></apply> </apply> </apply> </condition> </apply> </apply>
A third example shows the use of quantifiers with
condition
. The following markup encodes
"there exists
x < 3 such that
x^{2} = 4".
<apply> <exists/> <bvar><ci> x </ci></bvar> <condition> <apply><lt/><ci>x</ci><cn>3</cn></apply> </condition> <apply> <eq/> <apply> <power/><ci>x</ci><cn>2</cn> </apply> <cn>4</cn> </apply> </apply>
mappings 
semantics ,
annotation ,
annotationxml 
The use of content markup rather than presentation markup for mathematics is sometimes referred to as semantic tagging [Buswell1996]. The parsetree of a valid element structure using MathML content elements corresponds directly to the expression tree of the underlying mathematical expression. We therefore regard the content tagging itself as encoding the syntax of the mathematical expression. This is, in general, sufficient to obtain some rendering and even some symbolic manipulation (e.g. polynomial factorization).
However, even in such apparently simple expressions as
X +
Y, some additional information may be required for applications such as computer algebra. Are
X and
Y integers, or functions, etc.?
"Plus" represents addition over which field? This additional information is referred to as
semantic mapping. In MathML, this mapping is provided by the
semantics
,
annotation
and
annotationxml
elements.
The
semantics
element is the container element for the MathML expression together with its semantic mappings.
semantics
expects a variable number of child elements. The first is the element (which may itself be a complex element structure) for
which this additional semantic information is being defined. The second and subsequent children, if any, are instances of
the elements
annotation
and/or
annotationxml
.
The
semantics
element also accepts the
definitionURL
and
encoding
attributes for use by external processing applications. One use might be a URI for a semantic content dictionary, for example.
Since the semantic mapping information might in some cases be provided entirely by the
definitionURL
attribute, the
annotation
or
annotationxml
elements are optional.
The
annotation
element is a container for arbitrary data. This data may be in the form of text, computer algebra encodings, C programs,
or whatever a processing application expects.
annotation
has an attribute
"encoding" defining the form in use. Note that the content model of
annotation
is
PCDATA, so care must be taken that the particular encoding does not conflict with XML parsing rules.
The
annotationxml
element is a container for semantic information in wellformed XML. For example, an XML form of the OpenMath semantics could
be given. Another possible use here is to embed, for example, the presentation tag form of a construct given in content tag
form in the first child element of
semantics
(or vice versa).
annotationxml
has an attribute
"encoding" defining the form in use.
For example:
<semantics> <apply> <divide/> <cn>123</cn> <cn>456</cn> </apply> <annotation encoding="Mathematica"> N[123/456, 39] </annotation> <annotation encoding="TeX"> $0.269736842105263157894736842105263157894\ldots$ </annotation> <annotation encoding="Maple"> evalf(123/456, 39); </annotation> <annotationxml encoding="MathMLPresentation"> <mrow> <mn> 0.269736842105263157894 </mn> <mover accent='true'> <mn> 736842105263157894 </mn> <mo> ‾ </mo> </mover> </mrow> </annotationxml> <annotationxml encoding="OpenMath"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMS cd="arith1" name="divide"/> <OMI>123</OMI> <OMI>456</OMI> </OMA> </annotationxml> </semantics>
where
OMA
is the element defining the additional semantic information.
Of course, providing an explicit semantic mapping at all is optional, and in general would only be provided where there is some requirement to process or manipulate the underlying mathematics.
Although semantic mappings can easily be provided by various proprietary, or highly specialized encodings, there are no widely
available, nonproprietary standard schemes for semantic mapping. In part to address this need, the goal of the OpenMath effort
is to provide a platformindependent, vendorneutral standard for the exchange of mathematical objects between applications.
Such mathematical objects include semantic mapping information. The OpenMath group has defined an XML syntax for the encoding
of this information
[OpenMath2000]. This element set could provide the basis of one
annotationxml
element set.
An attractive side of this mechanism is that the OpenMath syntax is specified in XML, so that a MathML expression together with its semantic annotations can be validated using XML parsers.
MathML provides a collection of predefined constants and symbols which represent frequentlyencountered concepts in K12 mathematics.
These include symbols for wellknown sets, such as
integers
and
rationals
, and also some widely known constant symbols such as
false
,
true
,
exponentiale
.
MathML functions, operators and relations can all be thought of as mathematical functions if viewed in a sufficiently abstract way. For example, the standard addition operator can be regarded as a function mapping pairs of real numbers to real numbers. Similarly, a relation can be thought of as a function from some space of ordered pairs into the set of values {true, false}. To be mathematically meaningful, the domain and codomain of a function must be precisely specified. In practical terms, this means that functions only make sense when applied to certain kinds of operands. For example, thinking of the standard addition operator, it makes no sense to speak of "adding" a set to a function. Since MathML content markup seeks to encode mathematical expressions in a way that can be unambiguously evaluated, it is no surprise that the types of operands is an issue.
MathML specifies the types of arguments in two ways. The first way is by providing precise instructions for processing applications
about the kinds of arguments expected by the MathML content elements denoting functions, operators and relations. These operand
types are defined in a dictionary of default semantic bindings for content elements, which is given in
C Content Element Definitions. For example, the MathML content dictionary specifies that for real scalar arguments the plus operator is the standard commutative
addition operator over a field. The elements
cn
has a
type
attribute with a default value of
"real". Thus some processors will be able to use this information to verify the validity of the indicated operations.
Although MathML specifies the types of arguments for functions, operators and relations, and provides a mechanism for typing arguments, a MathMLcompliant processor is not required to do any type checking. In other words, a MathML processor will not generate errors if argument types are incorrect. If the processor is a computer algebra system, it may be unable to evaluate an expression, but no MathML error is generated.
Content element attributes are all of the type CDATA, that is, any character string will be accepted as valid. In addition, each attribute has a list of predefined values, which a content processor is expected to recognize and process. The reason that the attribute values are not formally restricted to the list of predefined values is to allow for extension. A processor encountering a value (not in the predefined list) which it does not recognize may validly process it as the default value for that attribute.
Each attribute is followed by the elements to which it can be applied.
base
indicates numerical base of the number. Predefined values: any numeric string.
The default value is "10"
closure
indicates closure of the interval. Predefined values: "open", "closed", "openclosed", "closedopen".
The default value is "closed"
definitionURL
points to an external definition of the semantics of the symbol or construct being declared. The value is a URL or URI that should point to some kind of definition. This definition overrides the MathML default semantics.
At present, MathML does not specify the format in which external semantic definitions should be given. In particular, there is no requirement that the target of the URI be loadable and parsable. An external definition could, for example, define the semantics in humanreadable form.
Ideally, in most situations the definition pointed to by the
definitionURL
attribute would be some standard, machinereadable format. However, there are reasons why MathML does not require such a
format.
No such format currently exists. There are several projects underway
to develop and implement standard semantic encoding formats,
most notably the OpenMath effort.
By nature, the development of a comprehensive system of semantic encoding
is a very large enterprise, and while much work has been done, much
additional work remains. Even though the
definitionURL
is designed and intended for use
with a formal semantic encoding language such as OpenMath, it is premature
to require any one particular format.
There will always be situations where some nonstandard format is preferable.
This is particularly true in situations where authors are describing new
ideas.
It is anticipated that in the near term, there will be a variety
of rendererdependent implementations of the
definitionURL
attribute.
A translation tool might simply prompt the user with the specified definition in situations where the proper semantics have been overridden, and in this case, humanreadable definitions will be most useful.
Other software may utilize OpenMath encodings.
Still other software may use proprietary encodings, or look for definitions in any of several formats.
As a consequence, authors need to be aware that there is no guarantee a generic renderer will be able to take advantage of
information pointed to by the
definitionURL
attribute. Of course, when widelyaccepted standardized semantic encodings are available, the definitions pointed to can
be replaced without modifying the original document. However, this is likely to be labor intensive.
There is no default value for the
definitionURL
attribute, i.e. the semantics are defined
within the MathML fragment, and/or by the MathML default semantics.
encoding
indicates the encoding of the annotation, or in the case of
csymbol
,
semantics
and operator elements, the syntax of the target referred to by
definitionURL
. Predefined values are
"MathMLPresentation",
"MathMLContent". Other typical values:
"TeX",
"OpenMath".
The default value is "", i.e. unspecified.
nargs
indicates number of arguments for function declarations. Predefined values: "nary", or any numeric string.
The default value is "1".
occurrence
indicates occurrence for operator declarations. Predefined values: "prefix", "infix", "functionmodel".
The default value is "functionmodel".
order
indicates ordering on the list. Predefined values: "lexicographic", "numeric".
The default value is "numeric".
scope
indicates scope of applicability of the declaration. Predefined values: "local", "global" (deprecated).
"local" means the containing MathML element.
"global" means the containing math
element.
In MathML 2.0, a declare has been restricted to occur only at the beginning of a
math
element. Thus, there is no difference between
the two possible scope
values and the scope attribute may be
safely ignored.
The "global" attribute value has been
deprecated for this role
as "local" better represents the concept.
Ideally, one would like to make documentwide declarations by setting the value of the
scope
attribute to be
"globaldocument". However, the proper mechanism for documentwide declarations very much depends on details of the way in
which XML will be embedded in HTML, future XML style sheet mechanisms, and the underlying Document Object Model.
Since these supporting technologies are still in flux at present, the MathML specification does not include
"globaldocument" as a predefined value of the
scope
attribute. It is anticipated, however, that this issue will be revisited in future revisions of MathML as supporting technologies
stabilize. In the near term, MathML implementors that wish to simulate the effect of a documentwide declaration are encouraged
to preprocess documents in order to distribute documentwide declarations to each individual
math
element in the document.
type
indicates type of the number. Predefined values: "enotation", "integer", "rational", "real", , "complexpolar", "complexcartesian", "constant".
The default value is "real".
Notes. Each data type implies that the data adheres to certain formatting conventions, detailed below. If the data fails to conform to the expected format, an error is generated. Details of the individual formats are:
A real number is presented in decimal notation. Decimal notation consists of an optional sign
("+" or
"") followed by a string of digits possibly separated into an integer and a fractional part by a
"decimal point". Some examples are 0.3, 1, and 31.56. If a different
base
is specified, then the digits are interpreted as being digits computed to that base.
A real number may also be presented in scientific notation. Such numbers have two parts (a mantissa and an exponent) separated
by
sep
. The first part is a real number, while the second part is an integer exponent indicating a power of the base.
For example, 12.3sep
5 represents
12.3 times 10^{5}.
The default presentation of this example is 12.3e5.
An integer is represented by an optional sign followed by a string of 1 or more
"digits". What a
"digit" is depends on the
base
attribute. If
base
is present, it specifies the base for the digit encoding, and it specifies it base 10. Thus
base
='16' specifies a hex encoding. When
base
> 10, letters are added in alphabetical order as digits. The legitimate values for
base
are therefore between 2 and 36.
A rational number is two integers separated by
sep
. If
base
is present, it specifies the base used for the digit encoding of both integers.
A complex number is of the form two real point numbers separated by
sep
.
A complex number is specified in the form of a magnitude and an angle (in radians). The raw data is in the form of two real
numbers separated by
sep
.
The
"constant" type is used to denote named constants. For example, an instance of
<cn type="constant">π</cn>
should be interpreted as having the semantics of the mathematical constant Pi. The data for a constant
cn
tag may be one of the following common constants:
Symbol  Value 
π 
The usual
π of trigonometry: approximately 3.141592653...

ⅇ (or
ⅇ )

The base for natural logarithms: approximately 2.718281828 ... 
ⅈ (or
ⅈ )

Square root of 1 
γ 
Euler's constant: approximately 0.5772156649... 
∞ (or
&infty; )

Infinity. Proper interpretation varies with context 
&true; 
the logical constant true 
&false; 
the logical constant false 
&NotANumber; (or
&NaN; )

represents the result of an illdefined floating point division 
indicates type of the identifier. Predefined values:
"integer",
"rational",
"real",
,
"complex",
"complexpolar",
"complexcartesian",
"constant", or the name of any content element. The meanings of the attribute values shared with
cn
are the same as those listed for the cn
element.
The attribute value "complex" is intended for use when an identifier
represents a complex number but the particular representation (such as polar or cartesian) is
either not known or is irrelevant.
The default value is "", i.e. unspecified.
indicates type of the identifier being declared. Predefined values: any content element name.
The default value is
ci
, i.e. a generic identifier
indicates type of the set. Predefined values: "normal", "multiset". "multiset" indicates that repetitions are allowed.
The default value is "normal".
is used to capture the notion of one quantity approaching another. It occurs as a container so that it can more easily be used in the construction of a limit expression. Predefined values: "above", "below", "twosided".
The default value is "above".
type
The
type
attribute, in addition to conveying semantic information, can be interpreted to provide rendering information. For example
in
<ci type="vector">V</ci>
a renderer could display a bold V for the vector.
All content elements support the following general attributes that can be used to modify the rendering of the markup.
class
style
id
other
The "class", "style" and "id" attributes are intended for compatibility with Cascading Style Sheets (CSS), as described in 2.4.5 Attributes Shared by all MathML Elements.
Content or semantic tagging goes along with the (frequently implicit) premise that, if you know the semantics, you can always work out a presentation form. When an author's main goal is to mark up reusable, evaluatable mathematical expressions, the exact rendering of the expression is probably not critical, provided that it is easily understandable. However, when an author's goal is more along the lines of providing enough additional semantic information to make a document more accessible by facilitating better visual rendering, voice rendering, or specialized processing, controlling the exact notation used becomes more of an issue.
MathML elements accept an attribute
other
(see
7.2.3 Attributes for unspecified data), which can be used to specify things not specifically documented in MathML. On content tags, this attribute can be used
by an author to express a
preference between equivalent forms for a particular content element construct, where the selection of the presentation has nothing
to do with the semantics. Examples might be
inline or displayed equations
scriptstyle fractions
use of x with a dot for a derivative over d x/d t
Thus, if a particular renderer recognized a display attribute to select between scriptstyle and displaystyle fractions, an author might write
<apply other='display="scriptstyle"'> <divide/> <cn> 1 </cn> <ci> x </ci> </apply>
to indicate that the rendering 1/ x is preferred.
The information provided in the
other
attribute is intended for use by specific renderers or processors, and therefore, the permitted values are determined by
the renderer being used. It is legal for a renderer to ignore this information. This might be intentional, as in the case
of a publisher imposing a house style, or simply because the renderer does not understand them, or is unable to carry them
out.
This section provides detailed descriptions of the MathML content tags. They are grouped in categories that broadly reflect the area of mathematics from which they come, and also the grouping in the MathML DTD. There is no linguistic difference in MathML between operators and functions. Their separation here and in the DTD is for reasons of historical usage.
When working with the content elements, it can be useful to keep in mind the following.
The role of the content elements is analogous to data entry in a mathematical system. The information that is provided is there to facilitate the successful parsing of an expression as the intended mathematical object by a receiving application.
MathML content elements do not by themselves "perform" any mathematical evaluations or operations. They do not "evaluate" in a browser and any "action" that is ultimately taken on those objects is determined entirely by the receiving mathematical application. For example, editing programs and applications geared to computation for the lower grades would typically leave 3 + 4 as is, whereas computational systems targeting a more advanced audience might evaluate this as 7. Similarly, some computational systems might evaluate sin(0) to 0, whereas others would leave it unevaluated. Yet other computational systems might be unable to deal with pure symbolic expressions like sin(x) and may even regard them as data entry errors. None of this has any bearing on the correctness of the original MathML representation. Where evaluation is mentioned at all in the descriptions below, it is merely to help clarify the meaning of the underlying operation.
Apart from the instances where there is an explicit interaction with presentation tagging, there is no required rendering (visual or aural)  only a suggested default. As such, the presentations that are included in this section are merely to help communicate to the reader the intended mathematical meaning by association with the same expression written in a more traditional notation.
The available content elements are:
token elements
basic content elements
arithmetic, algebra and logic
relations
calculus and vector calculus
theory of sets
sequences and series
elementary classical functions
statistics
linear algebra
vectorproduct
(MathML 2.0)
scalarproduct
(MathML 2.0)
outerproduct
(MathML 2.0)
semantic mapping elements
constant and symbol elements
integers
(MathML2.0)
reals
(MathML2.0)
rationals
(MathML2.0)
naturalnumbers
(MathML2.0)
complexes
(MathML2.0)
primes
(MathML2.0)
exponentiale
(MathML2.0)
imaginaryi
(MathML2.0)
notanumber
(MathML2.0)
true
(MathML2.0)
false
(MathML2.0)
emptyset
(MathML2.0)
pi
(MathML2.0)
eulergamma
(MathML2.0)
infinity
(MathML2.0)
cn
)
The cn
element is used to specify actual
numerical constants. The content model must provide sufficient information
that a number may be entered as data into a computational system. By
default, it represents a signed real number in base 10. Thus, the content
normally consists of PCDATA restricted to a sign, a string of
decimal digits and possibly a decimal point, or alternatively one of the
predefined symbolic constants such as π
.
The cn
element uses the attribute type
to represent other types of numbers such as, for
example, integer, rational, real or complex, and uses the attribute base
to specify the numerical base.
In addition to simple PCDATA, cn
accepts as content PCDATA separated by the (empty) element sep
. This determines the different parts needed to
construct a rational or complexcartesian number.
The cn
element may also contain arbitrary
presentation markup in its content (see 3 Presentation Markup) so that its
presentation can be very elaborate.
Alternative input notations for numbers are possible, but must be
explicitly defined by using the definitionURL
and
encoding
attributes, to refer to a written
specification of how a sequence of real numbers separated by sep
should be interpreted.
All attributes are CDATA:
Allowed values are "real", "integer", "rational", "complexcartesian", "complexpolar", "constant"
Number (CDATA for XML DTD) between 2 and 36.
URL or URI pointing to an alternative definition.
Syntax of the alternative definition.
<cn type="real"> 12345.7 </cn> <cn type="integer"> 12345 </cn> <cn type="integer" base="16"> AB3 </cn> <cn type="rational"> 12342 <sep/> 2342342 </cn> <cn type="complexcartesian"> 12.3 <sep/> 5 </cn> <cn type="complexpolar"> 2 <sep/> 3.1415 </cn> <cn type="constant"> π </cn>
By default, a contiguous block of
PCDATA contained in a
cn
element should render as if it were wrapped in an
mn
presentation element.
If an application supports bidirectional text rendering, then the
rendering within a cn
element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mn
presentation element.
Similarly, presentation markup contained in a
cn
element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were wrapped in an
mrow
element, with contiguous blocks of
PCDATA wrapped in
mn
elements.
However, not all mathematical systems that encounter content based tagging do visual or aural rendering. The receiving applications are free to make use of a number in the manner in which they normally handle numerical data. Some systems might simplify the rational number 12342/2342342 to 6171/1171171 while pure floating point based systems might approximate this as 0.5269085385e2. All numbers might be reexpressed in base 10. The role of MathML is simply to record enough information about the mathematical object and its structure so that it may be properly parsed.
The following renderings of the above MathML expressions are included both to help clarify the meaning of the corresponding MathML encoding and as suggestions for authors of rendering applications. In each case, no mathematical evaluation is intended or implied.
12345.7
12345
AB3_{16}
12342 / 2342342
12.3 + 5 i
Polar( 2 , 3.1415 )
ci
)
The
ci
element is used to name an identifier in a MathML expression (for example a variable). Such names are used to identify mathematical
objects. By default they are assumed to represent complex scalars. The
ci
element may contain arbitrary presentation markup in its content (see
3 Presentation Markup) so that its presentation as a symbol can be very elaborate.
The
ci
element uses the
type
attribute to specify the type of object that it represents. Valid types include
"integer",
"rational",
"real",
,
"complex",
"complexpolar",
"complexcartesian",
"constant", and more generally, any of the names of the MathML container elements (e.g.
vector
) or their type values. The
definitionURL
and
encoding
attributes can be used to extend the definition of
ci
to include other types. For example, a more advanced use might require a
"vector(complex)".
<ci> x </ci>
<ci type="vector"> V </ci>
<ci> <msub> <mi>x</mi> <mi>a</mi> </msub> </ci>
If the content of a
ci
element is tagged using presentation tags, that presentation is used. If no such tagging is supplied then the
PCDATA content would typically be rendered as if it were the content of an
mi
element.
If an application supports bidirectional text rendering, then the
rendering within a ci
element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mi
presentation element.
A renderer may wish to make use of the value of the type attribute to improve on this. For example, a symbol of type
vector
might be rendered using a bold face. Typical renderings of the above symbols are:
csymbol
)
The
csymbol
element allows a writer to create an element in MathML whose semantics are externally defined (i.e. not in the core MathML
content). The element can then be used in a MathML expression as for example an operator or constant. Attributes are used
to give the syntax and location of the external definition of the symbol semantics.
Use of
csymbol
for referencing external semantics can be contrasted with use of the
semantics
to attach additional information inline (ie. within the MathML fragment) to a MathML construct. See
4.2.6 Syntax and Semantics.
All attributes are CDATA:
Pointer to external definition of the semantics of the symbol. MathML does not specify a particular syntax in which this definition should be written.
Gives the syntax of the definition pointed to by definitionURL. An application can then test the value of this attribute to
determine whether it is able to process the target of the
definitionURL
. This syntax might be text, or a formal syntax such as OpenMath.
<! reference to OpenMath formal syntax definition of Bessel function > <apply> <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/BesselFunctions.ocd"> <msub><mi>J</mi><mn>0</mn></msub> </csymbol> <ci>y</ci> </apply> <! reference to human readable text description of Boltzmann's constant > <csymbol encoding="text" definitionURL="www.example.org/universalconstants/Boltzmann.htm"> k </csymbol>
By default, a contiguous block of
PCDATA contained in a
csymbol
element should render as if it were wrapped in an
mo
presentation element.
If an application supports bidirectional text rendering, then the
rendering within a csymbol
element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mo
presentation element.
Similarly, presentation markup contained in a
csymbol
element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrow
element, with contiguous blocks of
PCDATA wrapped in
mo
elements. The examples above would render by default as
As
csymbol
is used to support reference to externally defined semantics, it is a MathML error to have embedded content MathML elements
within the
csymbol
element.
apply
)
The
apply
element allows a function or operator to be applied to its arguments. Nearly all expression construction in MathML content
markup is carried out by applying operators or functions to arguments. The first child of
apply
is the operator to be applied, with the other child elements as arguments or qualifiers.
The
apply
element is conceptually necessary in order to distinguish between a function or operator, and an instance of its use. The
expression constructed by applying a function to 0 or more arguments is always an element from the codomain of the function.
Proper usage depends on the operator that is being applied. For example, the
plus
operator may have zero or more arguments, while the
minus
operator requires one or two arguments to be properly formed.
If the object being applied as a function is not already one of the elements known to be a function (such as
fn
,
sin
or
plus
) then it is treated as if it were the content of an
fn
element.
Some operators such as
diff
and
int
make use of
"named" arguments. These special arguments are elements that appear as children of the
apply
element and identify
"parameters" such as the variable of differentiation or the domain of integration. These elements are discussed further in
4.2.3.2 Operators taking Qualifiers.
<apply> <factorial/> <cn>3</cn> </apply>
<apply> <plus/> <cn>3</cn> <cn>4</cn> </apply>
<apply> <sin/> <ci>x</ci> </apply>
A mathematical system that has been passed an
apply
element is free to do with it whatever it normally does with such mathematical data. It may be that no rendering is involved
(e.g. a syntax validator), or that the
"function application" is evaluated and that only the result is rendered (e.g. sin(0)
0).
When an unevaluated
"function application" is rendered there are a wide variety of appropriate renderings. The choice often depends on the function
or operator being applied. Applications of basic operations such as
plus
are generally presented using an infix notation while applications of
sin
would use a more traditional functional notation such as sin(x). Consult the default rendering for the operator being applied.
Applications of userdefined functions (see
csymbol
,
fn
) that are not evaluated by the receiving or rendering application would typically render using a traditional functional notation
unless an alternative presentation is specified using the
semantics
tag.
reln
)
The
reln
element was used in MathML 1.0 to construct an equation or relation. Relations were constructed in a manner exactly analogous
to the use of
apply
. This usage is deprecated in MathML 2.0 in favor of the more generally usable
apply
.
The first child of
reln
is the relational operator to be applied, with the other child elements acting as arguments. See
4.2.4 Relations for further details.
fn
)
The
fn
element makes explicit the fact that a more general (possibly constructed) MathML object is being used in the same manner
as if it were a predefined function such as
sin
or
plus
.
fn
has exactly one child element, used to give the name (or presentation form) of the function. When
fn
is used as the first child of an apply, the number of following arguments is determined by the contents of the
fn
.
In MathML 1.0,
fn
was also the primary mechanism used to extend the collection of
"known" mathematical functions. This usage is now deprecated in favor of the more generally applicable
csymbol
element. (New functions may also be introduced by using
declare
in conjunction with a
lambda
expression.)
interval
)
The
interval
element is used to represent simple mathematical intervals of the real number line. It takes an attribute
closure
, which can take on any of the values
"open",
"closed",
"openclosed", or
"closedopen", with a default value of
"closed".
More general domains are constructed by using the
condition
and
bvar
elements to bind free variables to constraints.
The
interval
element expects
either two child elements that evaluate to real numbers
or one child element that is a
condition
defining the
interval
.
inverse
)
The
inverse
element is applied to a function in order to construct a generic expression for the functional inverse of that function.
(See also the discussion of
inverse
in
4.2.1.5 The inverse construct). As with other MathML functions,
inverse
may either be applied to arguments, or it may appear alone, in which case it represents an abstract inversion operator acting
on other functions.
A typical use of the
inverse
element is in an HTML document discussing a number of alternative definitions for a particular function so that there is
a need to write and define
f^{(1)}(x). To associate a particular definition with
f^{(1)}, use the
definitionURL
and
encoding
attributes.
sep
)
The sep
element is used to separate PCDATA
into separate tokens for parsing the contents of the various specialized
forms of the cn
elements. For example, sep
is used when specifying the real and imaginary
parts of a complex number (see 4.4.1 Token Elements). If it
occurs between MathML elements, it is a MathML error.
The sep
element is not directly rendered (see
4.4.1 Token Elements).
condition
)
The condition
element is used to place a
condition on one or more free variables or identifiers. The conditions may
be specified in terms of relations that are to be satisfied by the
variables, including general relationships such as set membership.
It is used to define general sets and lists in situations where the
elements cannot be explicitly enumerated. Condition contains either a
single apply
or reln
element; the apply
element
is used to construct compound conditions. For example, it is used below to
describe the set of all x such that x < 5. See the
discussion on sets in 4.4.6 Theory of Sets. See 4.2.5 Conditions for further details.
<condition> <apply><in/><ci> x </ci><ci type="set"> R </ci></apply> </condition>
<condition> <apply> <and/> <apply><gt/><ci> x </ci><cn> 0 </cn></apply> <apply><lt/><ci> x </ci><cn> 1 </cn></apply> </apply> </condition>
<apply> <max/> <bvar><ci> x </ci></bvar> <condition> <apply> <and/> <apply><gt/><ci> x </ci><cn> 0 </cn></apply> <apply><lt/><ci> x </ci><cn> 1 </cn></apply> </apply> </condition> <apply> <minus/> <ci> x </ci> <apply> <sin/> <ci> x </ci> </apply> </apply> </apply>
declare
)
The
declare
construct has two primary roles. The first is to change or set the default attribute values for a specific mathematical object.
The second is to establish an association between a
"name" and an object. Once a declaration is in effect, the
"name" object acquires the new attribute settings, and (if the second object is present) all the properties of the associated
object.
The various attributes of the
declare
element assign properties to the object being declared or determine where the declaration is in effect.
The scope of a declaration is "local" to the surrounding container element.
The scope
attribute can only be assigned to
"local", but is intended to support future extensions.
As discussed in
4.3.2.8
scope, MathML contains no provision for making documentwide declarations at present, though it is anticipated that this capability
will be added in future revisions of MathML, when supporting technologies become available.
declare
takes one or two children. The first child, which
is mandatory, is the object affected by the declaration.
This is usually a ci
element
providing the identifier that is being declared as in:
<declare type="vector"> <ci> V </ci> </declare>
The second child, which is optional, is a constructor initializing the variable:
<declare type="vector"> <ci> V </ci> <vector> <cn> 1 </cn><cn> 2 </cn><cn> 3 </cn> </vector> </declare>
The constructor type and the type of the element declared must agree. For example, if the type attribute of the declaration
is
fn
, the second child (constructor) must be an element equivalent to an
fn
element. (This would include actual
fn
elements,
lambda
elements and any of the defined functions in the basic set of content tags.) If no type is specified in the declaration then
the type attribute of the declared name is set to the type of the constructor (second child) of the declaration. The type
attribute of the declaration can be especially useful in the special case of the second element being a semantic tag.
All attributes are CDATA:
type
defines the MathML element type of the identifier declared.
scope
defines the scope of application of the declaration.
nargs
number of arguments for function declarations.
occurrence
describes operator usage as "prefix", "infix" or "functionmodel" indications.
definitionURL
URI pointing to detailed semantics of the function.
encoding
syntax of the detailed semantics of the function.
The declaration
<declare type="fn" nargs="2"> <ci> f </ci> <apply> <plus/> <ci> F </ci><ci> G </ci> </apply> </declare>
declares f to be a twovariable function with the property that f(x, y) = (F+ G)(x, y).
The declaration
<declare type="fn"> <ci> J </ci> <lambda> <bvar><ci> x </ci></bvar> <apply><ln/> <ci> x </ci> </apply> </lambda> </declare>
associates the name
J with a onevariable function defined so that
J(y) = ln
y. (Note that because of the type attribute of the
declare
element, the second argument must be something of function type
, namely a known function like
sin
, or a
lambda
construct.)
The
type
attribute on the declaration is only necessary if the type cannot be inferred from the type of the second argument.
Even when a declaration is in effect it is still possible to override attributes values selectively as in
<ci type="set"> S
</ci>
. This capability is needed in order to write statements of the form
"Let
s be a member of
S".
Since the
declare
construct is not directly rendered, most declarations are likely to be invisible to a reader. However, declarations can produce
quite different effects in an application which evaluates or manipulates MathML content. While the declaration
<declare> <ci> v </ci> <vector> <cn> 1 </cn> <cn> 2 </cn> <cn> 3 </cn> </vector> </declare>
is active the symbol v acquires all the properties of the vector, and even its dimension and components have meaningful values. This may affect how v is rendered by some applications, as well as how it is treated mathematically.
lambda
)
The lambda
element is used to construct a
userdefined function from an expression and one or more free
variables. The lambda construct with n internal variables takes
n+1 children. The first n children identify the variables
that are used as placeholders in the last child for actual parameter
values. See 4.2.2.2 Constructors for further details.
The first example presents a simple lambda construct.
<lambda> <bvar><ci> x </ci></bvar> <apply><sin/> <apply> <plus/> <ci> x </ci> <cn> 1 </cn> </apply> </apply> </lambda>
The next example constructs a oneargument function in which the argument b specifies the upper bound of a specific definite integral.
<lambda> <bvar><ci> b </ci></bvar> <apply> <int/> <bvar> <ci> x </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply><fn><ci> f </ci></fn> <ci> x </ci> </apply> </apply> </lambda>
Such constructs are often used in conjunction with
declare
to construct new functions.
compose
)
The compose
element represents the function
composition operator. Note that MathML makes no assumption about the domain
and codomain of the constituent functions in a composition; the domain of the
resulting composition may be empty.
To override the default semantics for the compose
element, or to associate a more specific
definition for function composition, use the definitionURL
and encoding
attributes. See 4.2.3 Functions, Operators and Qualifiers
for further details.
<apply> <compose/> <fn><ci> f </ci></fn> <fn><ci> g </ci></fn> </apply>
<apply> <compose/> <ci type="fn"> f </ci> <ci type="fn"> g </ci> <ci type="fn"> h </ci> </apply>
<apply> <apply><compose/> <fn><ci> f </ci></fn> <fn><ci> g </ci></fn> </apply> <ci> x </ci> </apply>
<apply> <fn><ci> f </ci></fn> <apply> <fn><ci> g </ci></fn> <ci> x </ci> </apply> </apply>
ident
)
The ident
element represents the identity
function. MathML makes no assumption about the function space in which the
identity function resides. That is, proper interpretation of the domain
(and hence codomain) of the identity function depends on the context in which
it is used.
To override the default semantics for the ident
element, or to associate a more specific
definition, use the definitionURL
and encoding
attributes (see 4.2.3 Functions, Operators and Qualifiers).
domain
)
The domain
element denotes the domain of a given function, which is the set of
values over which it is defined.
To override the default semantics for the
domain
element, or to associate a more specific
definition, use the definitionURL
and encoding
attributes (see 4.2.3 Functions, Operators and Qualifiers).
codomain
)
The codomain
element denotes the codomain of a given function, which is a set
containing all values taken by the function. It is not necessarily the case that every point in
the codomain is generated by the function applied to some point of the domain. (For example I may know
that a function is integervalued, so its codomain is the integers, without knowing (or stating) which
subset of the integers is mapped to by the function.)
Codomain is sometimes also called Range.
To override the default semantics for the
codomain
element, or to associate a more specific
definition, use the definitionURL
and encoding
attributes (see 4.2.3 Functions, Operators and Qualifiers).
image
)
The image
element denotes the image of a given function, which is the set
of values taken by the function. Every point in
the image is generated by the function applied to some point of the domain.
To override the default semantics for the
image
element, or to associate a more specific
definition, use the definitionURL
and encoding
attributes (see 4.2.3 Functions, Operators and Qualifiers).
domainofapplication
)
The domainofapplication
element denotes the domain over which a given function
is being applied. It is intended to be a more general alternative to specification of this
domain using such qualifier elements as bvar
, lowlimit
or condition
.
To override the default semantics for the
domainofapplication
element, or to associate a more specific
definition, use the definitionURL
and encoding
attributes (see 4.2.3 Functions, Operators and Qualifiers).
piecewise
, piece
,
otherwise
)
The
piecewise
,
piece
, and
otherwise
elements are used to support "piecewise" declarations of the form "
H(x) = 0 if x less than 0,
H(x) = 1 otherwise".
The declaration is constructed using the piecewise
element.
This contains one or more piece
elements, and optionally
one otherwise
element. Each piece
element contains exactly two children. The first child defines the value taken by the piecewise
expression when the condition specified in the associated second child of the piece
is true.
otherwise
allows the specification of a value to be taken by the
piecewise
function when none of the conditions (second child elements of the
piece
elements) is true, i.e. a default value.
It should be noted that no "order of execution" is implied by the ordering of the piece
child elements within piecewise
. It is the responsibility of the author
to ensure that the subsets of the function domain defined by the second children of the piece
elements are disjoint,
or that, where they overlap, the values of the corresponding first children of the piece
elements coincide. If this is not the case, the meaning of the expression is undefined.
The piecewise
elements are constructors
(see 4.2.2.2 Constructors).
<piecewise> <piece> <cn> 0 </cn> <apply><lt/><ci> x </ci> <cn> 0 </cn></apply> </piece> <otherwise> <ci> x </ci> </otherwise> </piecewise>
The following might be a definition of abs (x)
<apply> <eq/> <apply> <abs/> <ci> x </ci> </apply> <piecewise> <piece> <apply><minus/><ci> x </ci></apply> <apply><lt/><ci> x </ci> <cn> 0 </cn></apply> </piece> <piece> <cn> 0 </cn> <apply><eq/><ci> x </ci> <cn> 0 </cn></apply> </piece> <piece> <ci> x </ci> <apply><gt/><ci> x </ci> <cn> 0 </cn></apply> </piece> </piecewise> </apply>
quotient
)
The quotient
element is the operator used for
division modulo a particular base. When the quotient
operator is applied to integer arguments
a and b, the result is the "quotient of
a divided by b". That is, quotient
returns the unique integer q such
that a = q b + r. (In common usage,
q is called the quotient and r is the remainder.)
The quotient
element takes the attribute definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The quotient
element is a binary
arithmetic operator (see 4.2.3 Functions, Operators and Qualifiers).
factorial
)
The factorial
element is used to construct factorials.
The factorial
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The factorial
element is a
unary arithmetic operator (see
4.2.3 Functions, Operators and Qualifiers).
divide
)
The divide
element is the division operator.
The divide
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The divide
element is a
binary arithmetic operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <divide/> <ci> a </ci> <ci> b </ci> </apply>
As a MathML expression, this does not evaluate. However, on receiving such an expression, some applications may attempt to evaluate and simplify the value. For example, when a=5 and b=2 some mathematical applications may evaluate this to 2.5 while others will treat is as a rational number.
max
,
min
)
The elements
max
and
min
are used to compare the values of their arguments. They return the maximum and minimum of these values respectively.
The
max
and
min
elements take the
definitionURL
and
encoding
attributes that can be used to override the default semantics.
The
max
and
min
elements are
nary arithmetic operators (see
4.2.3 Functions, Operators and Qualifiers).
When the objects are to be compared explicitly they are listed as arguments to the function as in:
<apply> <max/> <ci> a </ci> <ci> b </ci> </apply>
The elements to be compared may also be described using bound variables with a
condition
element and an expression to be maximized (or minimized), as in:
<apply> <min/> <bvar><ci>x</ci></bvar> <condition> <apply><notin/><ci> x </ci><ci type="set"> B </ci></apply> </condition> <apply> <power/> <ci> x </ci> <cn> 2 </cn> </apply> </apply>
Note that the bound variable must be stated even if it might be implicit in conventional notation. In MathML1.0, the bound variable and expression to be evaluated (x) could be omitted in the example below: this usage is deprecated in MathML2.0 in favor of explicitly stating the bound variable and expression in all cases:
<apply> <max/> <bvar><ci>x</ci></bvar> <condition> <apply><and/> <apply><in/><ci>x</ci><ci type="set">B</ci></apply> <apply><notin/><ci>x</ci><ci type="set">C</ci></apply> </apply> </condition> <ci>x</ci> </apply>
minus
)
The minus
element is the subtraction operator.
The minus
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The minus
element can be used as a unary
arithmetic operator (e.g. to represent  x), or as a
binary arithmetic operator (e.g. to represent x
y).
plus
)
The plus
element is the addition operator.
The plus
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The plus
element is an nary arithmetic
operator (see 4.2.3 Functions, Operators and Qualifiers).
power
)
The power
element is a generic exponentiation
operator. That is, when applied to arguments a and b, it
returns the value of "a to the power of
b".
The
power
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
power
element is a
binary arithmetic operator (see
4.2.3 Functions, Operators and Qualifiers).
rem
)
The rem
element is the operator that returns the
"remainder" of a division modulo a particular base. When the
rem
operator is applied to integer arguments
a and b, the result is the "remainder of
a divided by b". That is, rem
returns the unique integer, r such that
a = q b+ r, where r <
q. (In common usage, q is called the quotient and
r is the remainder.)
The
rem
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
rem
element is a
binary arithmetic operator (see
4.2.3 Functions, Operators and Qualifiers).
times
)
The
times
element is the multiplication operator.
times
takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
root
)
The root
element is used to construct roots. The
kind of root to be taken is specified by a degree
element, which should be given as the second child
of the apply
element enclosing the root
element. Thus, square roots correspond to the case
where degree
contains the value 2, cube roots
correspond to 3, and so on. If no degree
is
present, a default value of 2 is used.
The root
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The root
element is an
operator taking qualifiers (see
4.2.3.2 Operators taking Qualifiers).
gcd
)
The gcd
element is used to denote the greatest
common divisor of its arguments.
The gcd
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The gcd
element is an
nary operator (see
4.2.3 Functions, Operators and Qualifiers).
and
)
The and
element is the boolean
"and" operator.
The
and
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
and
element is an
nary logical operator (see
4.2.3 Functions, Operators and Qualifiers).
or
)
The or
element is the boolean
"or" operator.
The
or
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
or
element is an
nary logical operator (see
4.2.3 Functions, Operators and Qualifiers).
xor
)
The xor
element is the boolean "exclusive
or" operator.
xor
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The xor
element is an nary logical
operator (see 4.2.3 Functions, Operators and Qualifiers).
not
)
The not
operator is the boolean
"not" operator.
The
not
element takes the attribute
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
not
element is a
unary logical operator (see
4.2.3 Functions, Operators and Qualifiers).
implies
)
The implies
element is the boolean relational operator
"implies".
The
implies
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
implies
element is a
binary logical operator (see
4.2.4 Relations).
forall
)
The
forall
element represents the universal quantifier of logic. It must be used in conjunction with one or more bound variables, an
optional
condition
element, and an assertion, which should take the form of an
apply
element. In MathML 1.0, the
reln
element was also permitted here: this usage is now deprecated.
The
forall
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
forall
element is a
quantifier (see
4.2.3.2 Operators taking Qualifiers).
The first example encodes a simple identity.
<apply> <forall/> <bvar><ci> x </ci></bvar> <apply><eq/> <apply> <minus/><ci> x </ci><ci> x </ci> </apply> <cn>0</cn> </apply> </apply>
The next example is more involved, and makes use of an optional
condition
element.
<apply> <forall/> <bvar><ci> p </ci></bvar> <bvar><ci> q </ci></bvar> <condition> <apply><and/> <apply><in/><ci> p </ci><rationals/></apply> <apply><in/><ci> q </ci><rationals/></apply> <apply><lt/><ci> p </ci><ci> q </ci></apply> </apply> </condition> <apply><lt/> <ci> p </ci> <apply> <power/> <ci> q </ci> <cn> 2 </cn> </apply> </apply> </apply>
The final example uses both the
forall
and
exists
quantifiers.
<apply> <forall/> <bvar><ci> n </ci></bvar> <condition> <apply><and/> <apply><gt/><ci> n </ci><cn> 0 </cn></apply> <apply><in/><ci> n </ci><integers/></apply> </apply> </condition> <apply> <exists/> <bvar><ci> x </ci></bvar> <bvar><ci> y </ci></bvar> <bvar><ci> z </ci></bvar> <condition> <apply><and/> <apply><in/><ci> x </ci><integers/></apply> <apply><in/><ci> y </ci><integers/></apply> <apply><in/><ci> z </ci><integers/></apply> </apply> </condition> <apply> <eq/> <apply> <plus/> <apply><power/><ci> x </ci><ci> n </ci></apply> <apply><power/><ci> y </ci><ci> n </ci></apply> </apply> <apply><power/><ci> z </ci><ci> n </ci></apply> </apply> </apply> </apply>
exists
)
The exists
element represents the existential
quantifier of logic. It must be used in conjunction with one or more bound
variables, an optional condition
element, and an
assertion, which may take the form of either an apply
or reln
element.
The exists
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The exists
element is a
quantifier (see 4.2.3.2 Operators taking Qualifiers).
abs
)
The abs
element represents the absolute value of
a real quantity or the modulus of a complex quantity.
The abs
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The abs
element is a unary arithmetic
operator (see 4.2.3 Functions, Operators and Qualifiers).
conjugate
)
The conjugate
element represents the complex
conjugate of a complex quantity.
The conjugate
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The conjugate
element is a unary
arithmetic operator (see 4.2.3 Functions, Operators and Qualifiers).
arg
)
The arg
operator (introduced in MathML 2.0)
gives the "argument" of a complex number, which is the angle
(in radians) it makes with the positive real axis. Real negative numbers
have argument equal to + .
The arg
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The arg
element is a unary arithmetic
operator (see 4.2.3 Functions, Operators and Qualifiers).
real
)
The real
operator (introduced in MathML 2.0)
gives the real part of a complex number, that is the x component in
x + i y
The real
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The real
element is a unary arithmetic
operator (see 4.2.3 Functions, Operators and Qualifiers).
imaginary
)
The imaginary
operator (introduced in MathML
2.0) gives the imaginary part of a complex number, that is, the y component
in x + i y.
The imaginary
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The imaginary
element is a unary
arithmetic operator (see 4.2.3 Functions, Operators and Qualifiers).
lcm
)
The
lcm
element (introduced in MathML 2.0) is used to denote the lowest common
multiple of its arguments.
The
lcm
takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
lcm
element is an
nary operator (see
4.2.3 Functions, Operators and Qualifiers).
floor
)
The
floor
element (introduced in MathML 2.0) is used to denote the
rounddown (towards infinity) operator.
The
floor
takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
floor
element is a
unary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <floor/> <ci> a </ci> </apply>
If this were evaluated at a = 15.015, it would yield 15.
<apply> <forall/> <bvar><ci> a </ci></bvar> <apply><and/> <apply><leq/> <apply><floor/> <ci>a</ci> </apply> <ci>a</ci> </apply> <apply><lt/> <ci>a</ci> <apply><plus/> <apply><floor/> <ci>a</ci> </apply> <cn>1</cn> </apply> </apply> </apply> </apply>
ceiling
)
The
ceiling
element (introduced in MathML 2.0) is used to denote the
roundup (towards +infinity) operator.
The
ceiling
takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
ceiling
element is a
unary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <ceiling/> <ci> a </ci> </apply>
If this were evaluated at a = 15.015, it would yield 16.
<apply> <forall/> <bvar><ci> a </ci></bvar> <apply><and/> <apply><lt/> <apply><minus/> <apply><ceiling/> <ci>a</ci> </apply> <cn>1</cn> </apply> <ci>a</ci> </apply> <apply><leq/> <ci>a</ci> <apply><ceiling/> <ci>a</ci> </apply> </apply> </apply> </apply>
eq
)
The eq
element is the relational operator
"equals".
The
eq
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
equals
element is an
nary relation (see
4.2.3.2 Operators taking Qualifiers).
neq
)
The neq
element is the "not equal
to" relational operator.
neq
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The neq
element is a binary
relation (see 4.2.4 Relations).
gt
)
The gt
element is the "greater
than" relational operator.
The gt
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The gt
element is an nary
relation (see 4.2.4 Relations).
lt
)
The lt
element is the "less than"
relational operator.
The lt
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The lt
element is an nary
relation (see 4.2.4 Relations).
geq
)
The geq
element is the relational operator
"greater than or equal".
The
geq
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
geq
element is an
nary relation (see
4.2.4 Relations).
leq
)
The leq
element is the relational operator
"less than or equal".
The
leq
element takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
The
leq
element is an
nary relation (see
4.2.4 Relations).
equivalent
)
The equivalent
element is the
"equivalence" relational operator.
The
equivalent
element takes the attributes
encoding
and definitionURL
that can be used to override the default semantics.
The
equivalent
element is an
nary relation (see
4.2.3.2 Operators taking Qualifiers).
approx
)
The approx
element is the relational operator
"approximately equal". This is a generic relational operator and no specific arithmetic precision is implied
The
approx
element takes the attributes
encoding
and definitionURL
that can be used to override the default semantics.
The
approx
element is a
binary relation (see
4.2.3.2 Operators taking Qualifiers).
factorof
)
The factorof
element is the relational operator
element on two integers a and b specifying whether
one is an integer factor of the other.
The factorof
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The factorof
element is an binary relational operator
(see 4.2.4 Relations).
int
)
The int
element is the operator element for an
integral. The lower limit, upper limit and bound variable are given by
(optional) child elements lowlimit
, uplimit
and bvar
in the
enclosing apply
element. The integrand is also
specified as a child element of the enclosing apply
element.
The domain of integration may be specified by using either an interval
element or a condition
element. In such cases, if a bound variable
of integration is intended, it must be specified explicitly. (The
condition may involve more than one symbol.)
The int
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The int
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
This example specifies a lowlimit
, uplimit
, and bvar
.
<apply> <int/> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn> 0 </cn> </lowlimit> <uplimit> <ci> a </ci> </uplimit> <apply> <ci> f </ci> <ci> x </ci> </apply> </apply>
This example specifies the domain of integration with an
interval
element.
<apply> <int/> <bvar> <ci> x </ci> </bvar> <interval> <ci> a </ci> <ci> b </ci> </interval> <apply><cos/> <ci> x </ci> </apply> </apply>
The final example specifies the domain of integration with a
condition
element.
<apply> <int/> <bvar> <ci> x </ci> </bvar> <condition> <apply><in/> <ci> x </ci> <ci type="set"> D </ci> </apply> </condition> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
diff
)
The diff
element is the differentiation operator
element for functions of a single variable. It may be applied directly to
an actual function such as sine or cosine, thereby denoting a function which is
the derivative of the original function, or it can be applied to an expression
involving a single variable such as sin(x), or cos(x). or a
polynomial in x. For the expression case the actual variable is
designated by a bvar
element that is a child of the
containing apply
element. The bvar
element may also contain a degree
element, which specifies the order of the
derivative to be taken.
The diff
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The diff
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
The derivative of a function f (often displayed as f') can be written as:
<apply> <diff/> <ci> f </ci> </apply>
The derivative with respect to x of an expression in x such as f (x) can be written as:
<apply> <diff/> <bvar> <ci> x </ci> </bvar> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
partialdiff
)
The partialdiff
element is the partial
differentiation operator element for functions or algebraic expressions in several
variables.
In the case of algebraic expressions, the bound variables are given by bvar
elements, which are children of the containing apply
element. The bvar
elements
may also contain degree
element, which specify
the order of the partial derivative to be taken in that variable.
For the expression case the actual variable is
designated by a bvar
element that is a child of the
containing apply
element. The bvar
elements may also contain a degree
element, which specifies the order of the
derivative to be taken.
Where a total degree of differentiation must be specified, this is indicated by use of a
degree
element at the top level, i.e. without any associated
bvar
, as a child
of the containing apply
element.
For the case of partial differentiation of a function, the containing apply
takes
two child elements: firstly a list of indices indicating by position
which coordinates are involved in
constructing the partial derivatives, and secondly the actual function to be partially differentiated.
The coordinates may be repeated.
The partialdiff
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The partialdiff
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
<apply><partialdiff/> <bvar><ci> x </ci><degree><ci> m </ci></degree></bvar> <bvar><ci> y </ci><degree><ci> n </ci></degree></bvar> <degree><ci> k </ci></degree> <apply><ci type="fn"> f </ci> <ci> x </ci> <ci> y </ci> </apply> </apply>
<apply><partialdiff/> <bvar><ci> x </ci></bvar> <bvar><ci> y </ci></bvar> <apply><ci type="fn"> f </ci> <ci> x </ci> <ci> y </ci> </apply> </apply>
<apply><partialdiff/> <list><cn>1</cn><cn>1</cn><cn>3</cn></list> <ci type="fn">f</ci> </apply>
lowlimit
)
The lowlimit
element is the container element
used to indicate the "lower limit" of an operator using
qualifiers. For example, in an integral, it can be used to specify the
lower limit of integration. Similarly, it can be used to specify the lower
limit of an index for a sum or product.
The meaning of the lowlimit
element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult 4.2.3.2 Operators taking Qualifiers.
<apply> <int/> <bvar> <ci> x </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
The default rendering of the
lowlimit
element and its contents depends on the context. In the preceding example, it should be rendered as a subscript to the integral
sign:
Consult the descriptions of individual operators that make use of the
lowlimit
construct for default renderings.
uplimit
)
The uplimit
element is the container element
used to indicate the "upper limit" of an operator using
qualifiers. For example, in an integral, it can be used to specify the
upper limit of integration. Similarly, it can be used to specify the upper
limit of an index for a sum or product.
The meaning of the uplimit
element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult 4.2.3.2 Operators taking Qualifiers.
<apply> <int/> <bvar> <ci> x </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
The default rendering of the uplimit
element and
its contents depends on the context. In the preceding example, it should be
rendered as a superscript to the integral sign:
Consult the descriptions of individual operators that make use of the
uplimit
construct for default renderings.
bvar
)
The bvar
element is the container element for
the "bound variable" of an operation. For example, in an
integral it specifies the variable of integration. In a derivative, it
indicates the variable with respect to which a function is being
differentiated. When the bvar
element is used to
qualify a derivative, the bvar
element may contain
a child degree
element that specifies the order of
the derivative with respect to that variable. The bvar
element is also used for the internal variable in
sums and products and for the bound variable used with the universal and
existential quantifiers forall
and exists
.
When a bvar
element has more than one
child element, the elements may appear in any order.
The meaning of the bvar
element depends on the
context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult 4.2.3.2 Operators taking Qualifiers.
<apply> <diff/> <bvar> <ci> x </ci> <degree> <cn> 2 </cn> </degree> </bvar> <apply> <power/> <ci> x </ci> <cn> 4 </cn> </apply> </apply>
<apply> <int/> <bvar><ci> x </ci></bvar> <condition> <apply><in/><ci> x </ci><ci> D </ci></apply> </condition> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
The default rendering of the
bvar
element and its contents depends on the context. In the preceding examples, it should be rendered as the
x in the dx of the integral, and as the
x in the denominator of the derivative symbol, respectively:
Note that in the case of the derivative, the default rendering of the
degree
child of the
bvar
element is as an exponent.
Consult the descriptions of individual operators that make use of the
bvar
construct for default renderings.
degree
)
The degree
element is the container element for
the "degree" or "order" of an operation. There
are a number of basic mathematical constructs that come in families, such as
derivatives and moments. Rather than introduce special elements for each of
these families, MathML uses a single general construct, the degree
element for this concept of
"order".
The meaning of the
degree
element depends on the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking qualifiers, consult
4.2.3.2 Operators taking Qualifiers.
<apply> <partialdiff/> <bvar> <ci> x </ci> <degree> <ci> n </ci> </degree> </bvar> <bvar> <ci> y </ci> <degree> <ci> m </ci> </degree> </bvar> <apply><sin/> <apply> <times/> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply>
The default rendering of the
degree
element and its contents depends on the context. In the preceding example, the
degree
elements would be rendered as the exponents in the differentiation symbols:
Consult the descriptions of individual operators that make use of the
degree
construct for default renderings.
divergence
)
The divergence
element is the vector calculus
divergence operator, often called div.
The divergence
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The divergence
element is a
unary calculus operator (see
4.2.3 Functions, Operators and Qualifiers).
grad
)
The grad
element is the vector calculus gradient
operator, often called grad.
The grad
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The grad
element is a unary calculus
operator (see 4.2.3 Functions, Operators and Qualifiers).
curl
)
The curl
element is the vector calculus curl operator.
The curl
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The curl
element is a unary calculus
operator (see 4.2.3 Functions, Operators and Qualifiers).
laplacian
)
The laplacian
element is the vector calculus
laplacian operator.
The laplacian
element takes the attributes encoding
and definitionURL
that can be used to override the
default semantics.
The laplacian
element is an unary calculus
operator (see 4.2.3 Functions, Operators and Qualifiers).
set
)
The
set
element is the container element that constructs a set of elements. The elements of a set can be defined either by explicitly
listing the elements, or by using the
bvar
and
condition
elements.
The
set
element is a
constructor element (see
4.2.2.2 Constructors).
<set> <ci> b </ci> <ci> a </ci> <ci> c </ci> </set>
This constructs the set {b, a, c}
<set> <bvar><ci> x </ci></bvar> <condition> <apply><and/> <apply><lt/> <ci> x </ci> <cn> 5 </cn> </apply> <apply><in/> <ci> x </ci> <naturalnumbers/> </apply> </apply> </condition> <ci> x </ci> </set>
This constructs the set of all natural numbers less than 5, ie. the set {0, 1, 2, 3, 4}
list
)
The
list
element is the container element that constructs a list of elements. Elements can be defined either by explicitly listing
the elements, or by using the
bvar
and
condition
elements.
Lists differ from sets in that there is an explicit order to the elements. Two orders are supported: lexicographic and numeric.
The kind of ordering that should be used is specified by the
order
attribute.
The
list
element is a
constructor element (see
4.2.2.2 Constructors).
union
)
The union
element is the operator for a
settheoretic union or join of two (or more) sets.
The union
attribute takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The union
element is an nary set
operator (see 4.2.3 Functions, Operators and Qualifiers).
intersect
)
The intersect
element is the operator for the
settheoretic intersection or meet of two (or more) sets.
The intersect
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The intersect
element is an nary set
operator (see 4.2.3 Functions, Operators and Qualifiers).
in
)
The in
element is the relational operator used
for a settheoretic inclusion ("is in" or "is a member
of").
The in
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The in
element is a binary set
relation (see 4.2.4 Relations).
notin
)
The notin
element is the relational operator
element used for settheoretic exclusion ("is not in" or
"is not a member of").
The notin
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The notin
element is a binary set
relation (see 4.2.4 Relations).
subset
)
The subset
element is the relational operator
element for a settheoretic containment ("is a subset
of").
The subset
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The subset
element is an nary set
relation (see 4.2.4 Relations).
prsubset
)
The prsubset
element is the relational operator
element for settheoretic proper containment ("is a proper subset
of").
The prsubset
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The subset
element is an nary set
relation (see 4.2.4 Relations).
notsubset
)
The notsubset
element is the relational operator
element for the settheoretic relation "is not a subset
of".
The notsubset
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The notsubset
element is a binary set
relation (see 4.2.4 Relations).
notprsubset
)
The notprsubset
element is the operator element
for the settheoretic relation "is not a proper subset
of".
The notprsubset
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The notprsubset
element is a binary set
relation (see 4.2.4 Relations).
setdiff
)
The setdiff
element is the operator element for
a settheoretic difference of two sets.
The setdiff
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The setdiff
element is a binary set
operator (see 4.2.3 Functions, Operators and Qualifiers).
card
)
The card
element is the operator element for
the size or cardinality of a set.
The card
element takes the attributes definitionURL
and encoding
that can be used to override the
default semantics.
The card
element is a unary set
operator (see 4.2.3 Functions, Operators and Qualifiers).
cartesianproduct
)
The cartesianproduct
element is the operator element for
the Cartesian product of two or more sets. If A and B are two sets, then
the Cartesian product of A and B is the set of all pairs (a,b)
with a in A and b in B.
The cartesianproduct
element takes the attributes definitionURL
and encoding
that can be used to override the
default semantics.
The cartesianproduct
element is a nary set
operator (see 4.2.3 Functions, Operators and Qualifiers).
sum
)
The sum
element denotes the summation
operator. Upper and lower limits for the index of a sum can be specified using
uplimit
and lowlimit
. More general
domains for the indices can be specified using a condition
involving the bound variables. The index for the summation is specified by a
bvar
element.
The sum
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The sum
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
<apply> <sum/> <bvar> <ci> x </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply> <apply> <sum/> <bvar> <ci> x </ci> </bvar> <condition> <apply> <in/> <ci> x </ci> <ci type="set"> B </ci> </apply> </condition> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
product
)
The product
element denotes the product
operator. Upper and lower limits for the index of a product can be specified using
uplimit
and lowlimit
. More general
domains for the indices can be specified using a condition
involving the bound variables. The index for the product is specified by a
bvar
element.
The product
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The product
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
<apply> <product/> <bvar> <ci> x </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply> <apply> <product/> <bvar> <ci> x </ci> </bvar> <condition> <apply> <in/> <ci> x </ci> <ci type="set"> B </ci> </apply> </condition> <apply><ci type="fn"> f </ci> <ci> x </ci> </apply> </apply>
limit
)
The limit
element represents the operation of
taking a limit of a sequence. The limit point is expressed by specifying a
lowlimit
and a bvar
, or by
specifying a condition
on one or more bound
variables.
The limit
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The limit
element is an operator taking
qualifiers (see 4.2.3.2 Operators taking Qualifiers).
<apply> <limit/> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn> 0 </cn> </lowlimit> <apply><sin/> <ci> x </ci> </apply> </apply>
<apply> <limit/> <bvar> <ci> x </ci> </bvar> <condition> <apply> <tendsto type="above"/> <ci> x </ci> <ci> a </ci> </apply> </condition> <apply><sin/> <ci> x </ci> </apply> </apply>
tendsto
)
The tendsto
element is used to express the
relation that a quantity is tending to a specified value.
The tendsto
element takes the attributes type
to set the direction from which the limiting
value is approached.
The tendsto
element is a binary relational
operator (see 4.2.4 Relations).
<apply> <tendsto type="above"/> <apply> <power/> <ci> x </ci> <cn> 2 </cn> </apply> <apply> <power/> <ci> a </ci> <cn> 2 </cn> </apply> </apply>
To express (x, y) (f(x, y), g(x, y)), one might use vectors, as in:
<apply> <tendsto/> <vector> <ci> x </ci> <ci> y </ci> </vector> <vector> <apply><ci type="fn"> f </ci> <ci> x </ci> <ci> y </ci> </apply> <apply><ci type="fn"> g </ci> <ci> x </ci> <ci> y </ci> </apply> </vector> </apply>
The names of the common trigonometric functions supported by MathML are listed below. Since their standard interpretations are widely known, they are discussed as a group.
sin 
cos 
tan 
sec 
csc 
cot 
sinh 
cosh 
tanh 
sech 
csch 
coth 
arcsin 
arccos 
arctan 
arccosh 
arccot 
arccoth 
arccsc 
arccsch 
arcsec 
arcsech 
arcsinh 
arctanh 
These operator elements denote the standard trigonometrical functions.
These elements all take the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
They are all unary trigonometric operators. (see 4.2.3 Functions, Operators and Qualifiers).
exp
)
The exp
element represents the exponential
function associated with the inverse of the ln
function. In particular, exp(1) is approximately 2.718281828.
The exp
element takes the definitionURL
and encoding
attributes, which may be used to override the
default semantics.
The exp
element is a unary arithmetic
operator (see 4.2.3 Functions, Operators and Qualifiers).
ln
)
The ln
element represents the natural logarithm
function.
The ln
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The ln
element is a unary calculus
operator (see 4.2.3 Functions, Operators and Qualifiers).
log
)
The log
element is the operator that returns a
logarithm to a given base. The base may be specified using a logbase
element, which should be the first element
following log
, i.e. the second child of the
containing apply
element. If the logbase
element is not present, a default base of 10 is
assumed.
The log
element takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
The log
element can be used as either an
operator taking qualifiers or a unary calculus
operator (see 4.2.3.2 Operators taking Qualifiers).
mean
)
mean
is the operator element representing a mean
or average.
mean
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
mean
is an nary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <mean/> <ci> X </ci> </apply>
sdev
)
sdev
is the operator element representing the
statistical standard deviation operator.
sdev
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
sdev
is an nary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <sdev/> <ci> X </ci> </apply>
variance
)
variance
is the operator element representing the
statistical variance operator.
variance
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
variance
is an nary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <variance/> <ci> X </ci> </apply>
median
)
median
is the operator element representing the statistical
median operator.
median
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
median
is an nary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <median/> <ci> X </ci> </apply>
mode
)
mode
is the operator element representing the statistical
mode operator.
mode
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
mode
is an nary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <mode/> <ci> X </ci> </apply>
moment
)
The moment
element represents the statistical
moment operator. Use the qualifier degree
for the n in
" nth moment". Use the qualifier momentabout
for the p in
"moment about p".
moment
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
moment
is an operator taking qualifiers (see
4.2.3.2 Operators taking Qualifiers). The third moment of the distribution
X about the point p is written:
<apply> <moment/> <degree> <cn> 3 </cn> </degree> <momentabout> <ci> p </ci> </momentabout> <ci> X </ci> </apply>
momentabout
)
The momentabout
element is a qualifier element used with the
moment
element to represent statistical
moments. Use the qualifier momentabout
for the p in
"moment about p".
momentabout
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
vector
)
vector
is the container element for a
vector. The child elements form the components of the vector.
For purposes of interaction with matrices and matrix multiplication, vectors are regarded as equivalent to a matrix consisting of a single column, and the transpose of a vector behaves the same as a matrix consisting of a single row.
vector
is a constructor element (see
4.2.2.2 Constructors).
<vector> <cn> 1 </cn> <cn> 2 </cn> <cn> 3 </cn> <ci> x </ci> </vector>
matrix
)
The matrix
element is the container element for
matrix rows, which are represented by matrixrow
. The matrixrow
s
contain the elements of a matrix.
matrix
is a constructor element (see
4.2.2.2 Constructors).
<matrix> <matrixrow> <cn> 0 </cn> <cn> 1 </cn> <cn> 0 </cn> </matrixrow> <matrixrow> <cn> 0 </cn> <cn> 0 </cn> <cn> 1 </cn> </matrixrow> <matrixrow> <cn> 1 </cn> <cn> 0 </cn> <cn> 0 </cn> </matrixrow> </matrix>
matrixrow
)
matrixrow
is a constructor element (see
4.2.2.2 Constructors).
<matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow> <matrixrow> <cn> 3 </cn> <ci> x </ci> </matrixrow>
determinant
)
The determinant
element is the operator for constructing the determinant of a matrix.
determinant
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
determinant
is a unary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <determinant/> <ci type="matrix"> A </ci> </apply>
transpose
)
The transpose
element is the operator for
constructing the transpose of a matrix.
transpose
takes the definitionURL
and encoding
attributes, which can be used to override the
default semantics.
transpose
is a
unary operator (see
4.2.3 Functions, Operators and Qualifiers).
<apply> <transpose/> <ci type="matrix"> A </ci> </apply>
selector
)
The selector
element is the operator for
indexing into vectors matrices and lists. It accepts one or more
arguments. The first argument identifies the vector, matrix or list from
which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.
When selector
is used with a single argument, it
should be interpreted as giving the sequence of all elements in the list,
vector or matrix given. The ordering of elements in the sequence for a
matrix is understood to be first by column, then by row. That is, for a
matrix ( a_{i,j}), where the indices
denote row and column, the ordering would be a _{1,1},
a _{1,2}, ... a _{2,1}, a_{2,2} ... etc.
When three arguments are given, the last one is ignored for a list or vector, and in the case of a matrix, the second and third arguments specify the row and column of the selected element.
When two arguments are given, and the first is a vector or list, the second argument specifies an element in the list or vector. When a matrix and only one index i is specified as in
<apply> <selector/> <matrix> <matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow> <matrixrow> <cn> 3 </cn> <cn> 4 </cn> </matrixrow> </matrix> <cn> 1 </cn> </apply>
it refers to the ith matrixrow. Thus, the preceding example selects the following row:
<matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow>
selector
takes the
definitionURL
and
encoding
attributes, which can be used to override the default semantics.
selector
is classified as an nary linear algebra operator even though it can take only one, two, or three arguments.
<apply> <selector/> <ci type="matrix"> A </ci> <cn> 3 </cn> <cn> 2 </cn> </apply>
The selector
construct renders in a manner that indicates
which subelement of the parent object is selected. For vectors and matrices this is
normally done by specifying the parent object together with subscripted indices.
For example, the selection
<apply> <selector/> <ci type="vector">V</ci> <cn> 1 </cn> </apply>
would have a default rendering of
Selecting the (1,2) element of a 2 by 2 matrix would have a default rendering as
vectorproduct
)
The vectorproduct
is the operator element for
deriving the vector product of two vectors.
The vectorproduct
element takes the attributes
definitionURL
and encoding
that can be used to override
the default semantics.
The vectorproduct
element is a binary
vector operator (see 4.2.3 Functions, Operators and Qualifiers).
<apply> <eq/> <apply><vectorproduct/> <ci type="vector"> A </ci> <ci type="vector"> B </ci> </apply> <apply><times/> <ci> a </ci> <ci> b </ci> <apply><sin/> <ci> θ </ci> </apply> <ci type="vector"> N </ci> </apply> </apply>
where A and B are vectors, N is a unit vector orthogonal to A and B, a, b are the magnitudes of A, B and is the angle between A and B.
scalarproduct
)
The scalarproduct
is the operator element for
deriving the scalar product of two vectors.
The scalarproduct
element takes the attributes
definitionURL
and encoding
that can be used to override
the default semantics.
The scalarproduct
element is a binary
vector operator (see 4.2.3 Functions, Operators and Qualifiers).
<apply> <eq/> <apply><scalarproduct/> <ci type="vector"> A </ci> <ci type="vector">B </ci> </apply> <apply><times/> <ci> a </ci> <ci> b </ci> <apply><cos/> <ci> θ </ci> </apply> </apply> </apply>
where A and B are vectors, a, b are the magnitudes of A, B and is the angle between A and B.
outerproduct
)
The outerproduct
is the operator element for
deriving the outer product of two vectors.
The outerproduct
element takes the attributes
definitionURL
and encoding
that can be used to override
the default semantics.
The outerproduct
element is a
binary vector operator (see 4.2.3 Functions, Operators and Qualifiers).
This section explains the use of the semantic mapping elements semantics
, annotation
and annotationxml
.
annotation
)
The annotation
element is the container element
for a semantic annotation in a nonXML format.
The annotation
element takes the attribute encoding
to define the encoding being used.
semantics
)
The semantics
element is the container element
that associates additional representations with a given MathML
construct. The semantics
element has as its first
child the expression being annotated, and the subsequent children are the
annotations. There is no restriction on the kind of annotation that can be
attached using the semantics element. For example, one might give a T_{E}X
encoding, or computer algebra input in an annotation.
The representations that are XML based are enclosed in an annotationxml
element while those representations that
are to be parsed as PCDATA are enclosed in an annotation
element.
The semantics
element takes the definitionURL
and encoding
attributes, which can be used to reference an
external source for some or all of the semantic information.
An important purpose of the semantics
construct
is to associate specific semantics with a particular presentation, or
additional presentation information with a content construct. The default
rendering of a semantics
element is the default
rendering of its first child. When a MathMLpresentation annotation is
provided, a MathML renderer may optionally use this information to render
the MathML construct. This would typically be the case when the first child
is a MathML content construct and the annotation is provided to give a
preferred rendering differing from the default for the content
elements.
Use of semantics
to attach additional
information inline to a MathML construct can be contrasted with use of the
csymbol
for referencing external semantics. See
4.4.1.3 Externally defined symbol (csymbol)
The semantics
element is a semantic mapping element.
<semantics> <apply> <plus/> <apply> <sin/> <ci> x </ci> </apply> <cn> 5 </cn> </apply> <annotation encoding="Maple"> sin(x) + 5 </annotation> <annotationxml encoding="MathMLPresentation"> ... ... </annotationxml> <annotation encoding="Mathematica"> Sin[x] + 5 </annotation> <annotation encoding="TeX"> \sin x + 5 </annotation> <annotationxml encoding="OpenMath"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMS cd="transc1" name="sin"/> <OMI>5</OMI> </OMA> </annotationxml> </semantics>
annotationxml
)
The annotationxml
container element is used to
contain representations that are XML based. It is always used together with
the semantics
element, and takes the attribute encoding
to define the encoding being used.
annotationxml
is a semantic mapping element.
<semantics> <apply> <plus/> <apply><sin/> <ci> x </ci> </apply> <cn> 5 </cn> </apply> <annotationxml encoding="OpenMath"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMS name="plus" cd="arith1"/> <OMA><OMS name="sin" cd="transc1"/> <OMV name="x"/> </OMA> <OMI>5</OMI> </OMA> </annotationxml> </semantics>
See also the discussion of semantics
above.
This section explains the use of the Constant and Symbol elements.
naturalnumbers
)
complexes
)
primes
)
exponentiale
)
imaginaryi
)
imaginaryi
represents the mathematical constant
which is the square root of 1, commonly written i.
notanumber
)
notanumber
represents the result of an
illdefined floating point operation, sometimes also called
NaN.
true
)
false
)
pi
)