W3C MathML TestSuite: Version 3: Test-case complex4

Alternatives:	(mml file) (full) (simple) (plain) (form) (slideshow)
File:	TortureTests/Complexity/complex4
CVS-ID:
Author:	Design Science, Inc. (D. Doyle, R. Miner)
Description:	Testing content tags.

passed (p) failed (f) not-tested (n) broken-test (b) some parts pass (s)

Sample Rendering: N/A
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a + b - c

$a b c$
x + yz + z

$x y z z$
x * (y + z) * z

$x y z z$
sin(a + b)

$a b$
sin(x(y + z)z)

$x y z z$
sin(xy)2b

$x y 2 b$
(x + y) ^ (n - 3)

$x y n 3$
limit as x goes to a of sin x using <reln>

$x x a x$
limit as x goes to a of sin x using <apply>

$x x a x$
limit as x goes to a of sin(x + y) using <reln>

$x x a x y$
limit as x goes to a of sin(x + y) using <apply>

$x x a x y$
limit as x goes to a of sin(x + y)2b using <reln>

$x x a x y 2 b$
limit as x goes to a of sin(x + y)2b using <apply>

$x x a x y 2 b$
quotient

$a b$
moment

$3 X$
selector

$12341$
factorial

$n$
(a + b)!

$n m x$
inverse function

$f$
inverse matrix

$a A$
conjugate

$x ⅈ y$
a + b + c

$a b c$
integral (a + x)dx

$x 0 a a x$
-1 + 7

$17$
7 + (-1)

$71$
max

$x x 0 x 1 x x$
lambda sin(x + 1)

$x x 1$
lambda integral f(x)dx

$b x a b f x$
compose f and g

$f g$
compose f and g (x)

$f g x$
f(g(x))

$f g x$
composition of f and inverse of f eq identity using <reln>

$f f$
composition of f and inverse of f eq identity using <apply>

$f f$
e^x

$x$
min(x, x not in B, x^2) using <reln>

$x x B x 2$
min(x, x not in B, x^2) using <apply>

$x x B x 2$
a mod (b)

$a b$
ab

$a b$
gcd(a b, c)

$a b c$
integral

$x 0 a f x$
abs(x)

$x$
tall abs(x)

$H K$
abs(x + y + z)

$x y z$
x > 0 and z < 1 as <reln>

$x 0 x 1$
x > 0 and z < 1 as <apply>

$x 0 x 1$
a and b

$a b$
a or b

$a b$
a xor b

$a b$
a eq b with <reln> tag

$a b$
a eq b with <apply> tag

$a b$
a neq b with <reln> tag

$a b$
a neq b with <apply> tag

$a b$
a > b with <reln> tag

$a b$
a > b with <apply> tag

$a b$
a < b with <reln> tag

$a b$
a < b with <apply> tag

$a b$
a <= b with <reln> tag

$a b$
a >= b with <apply> tag

$a b$
a <= b with <reln> tag

$a b$
a <= b with <apply> tag

$a b$
set: {b, a, c}

$b a c$
set with condition

$x x 5$
list: {b, a, c}

$b a c$
list: {x|x < 5}

$x x 5$
A union B

$A B$
A intersect B

$A B$
A intersect (B union C)

$A B C$
integral x in R as <reln>

$x R$
x in R as <apply>

$x R$
a in A

$a A$
a not in A as <reln>

$a A$
a not in A as <apply>

$a A$
not a

$a$
not (a and b)

$a b$
A -> B (reln)

$A B$
A -> B (apply)

$A B$
forall

$x x x 0$
forall/and/lt/power

$p q p Q q Q p q p q 2$
forall/exists/and/plus

$n n Z n 0 x y z x Z y Z z Z x n y n z n$
exists

$x f x 0$
forall/exists/and/plus

$n n Z n 0 x y z x Z y Z z Z x n y n z n$
ln a

$a$
log base 3 of x

$3 x$
integer

$x x D f x$
diff

$x f x$
partialdiff

$x 2 y f x y$
integral

$x a b f x$
partialdiff

$x n y m x y$
divide

$a b$
divide/plus/minus

$a b a b$
divide/plus/divide

$a b a b$
A is subset of B as <reln>

$A B$
A is subset of B as <apply>

$A B$
A is proper subset of B as <reln>

$A B$
A is proper subset of B as <apply>

$A B$
A is not subset of B as <reln>

$A B$
A is not subset of B as <apply>

$A B$
A is not proper subset of B as <reln>

$A B$
A is not proper subset of B as <apply>

$A B$
Set difference

$A B$
Log base 3 of x + y

$3 x y$
Sum as x goes from a to b of f(x)

$x a b f x$
sum

$x x B f x$
product

$x a b f x$
product

$x x B f x$
tendsto with <reln>

$x 2 a 2$
tendsto with <apply>

$x 2 a 2$
tendsto with <reln>

$x y f x y g x y$
tendsto with <apply>

$x y f x y g x y$
mean(X)

$X$
root(a + b)

$n a b$
standard deviation

$X$
variance(X)

$X$
median(X)

$X$
mode(X)

$X$
degree

$3 X$
vector

$123 x$
matrix

$010000101000$
determinant

$A$
transpose

$A$
semantics

$\sin x + 5$
limit

$x 0 x$
symbol check

$4.56 4.56 4 5 4 5 4.56 4.56 π ⅇ ⅇ ⅈ ⅈ γ ∞ ∞$
multiset

$4.56 4.56 4 5 4 5 4.56 4.56 π ⅇ ⅇ ⅈ ⅈ γ ∞ ∞$
tendsto type = "above" with <reln>

$x 2 a 2$
tendsto type = "above" with <apply>

$x 2 a 2$
tendsto type = "below" with <reln>

$x 2 a 2$
tendsto type = "below" with <apply>

$x 2 a 2$
tendsto type = "two-sided" with <reln>

$x 2 a 2$
tendsto type = "two-sided" with <apply>

$x 2 a 2$
type check

$x x x y θ v π ⅇ ⅇ ⅈ ⅈ γ ∞ ∞$
sin + cos

$x x$
f(x)

$f x$
Source Code:

    <p>a + b - c 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <ci>a</ci> 
          <apply> 
            <minus/> 
            <ci>b</ci> 
            <ci>c</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
x + yz + z 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <ci>x</ci> 
          <apply> 
            <times/> 
            <ci>y</ci> 
            <ci>z</ci> 
          </apply> 
          <ci>z</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
x * (y + z) * z 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <times/> 
          <ci>x</ci> 
          <apply> 
            <plus/> 
            <ci>y</ci> 
            <ci>z</ci> 
          </apply> 
          <ci>z</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
sin(a + b) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sin/> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
sin(x(y + z)z) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sin/> 
          <apply> 
            <times/> 
            <ci>x</ci> 
            <apply> 
              <plus/> 
              <ci>y</ci> 
              <ci>z</ci> 
            </apply> 
            <ci>z</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
sin(xy)2b 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <times/> 
          <apply> 
            <sin/> 
            <apply> 
              <times/> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
          </apply> 
          <apply> 
            <times/> 
            <cn>2</cn> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
(x + y) ^ (n - 3) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <power/> 
          <apply> 
            <plus/> 
            <ci>x</ci> 
            <ci>y</ci> 
          </apply> 
          <apply> 
            <minus/> 
            <cn>n</cn> 
            <cn>3</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin x using &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <tendsto type='above'/> 
              <ci>x</ci> 
              <ci>a</ci> 
            </reln> 
          </condition> 
          <apply> 
            <sin/> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin x using &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <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> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin(x + y) using &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <tendsto type='above'/> 
              <ci>x</ci> 
              <ci>a</ci> 
            </reln> 
          </condition> 
          <apply> 
            <sin/> 
            <apply> 
              <plus/> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin(x + y) using &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <tendsto type='above'/> 
              <ci>x</ci> 
              <ci>a</ci> 
            </apply> 
          </condition> 
          <apply> 
            <sin/> 
            <apply> 
              <plus/> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin(x + y)2b using &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <tendsto type='above'/> 
              <ci>x</ci> 
              <ci>a</ci> 
            </reln> 
          </condition> 
          <apply> 
            <times/> 
            <apply> 
              <sin/> 
              <apply> 
                <times/> 
                <ci>x</ci> 
                <ci>y</ci> 
              </apply> 
            </apply> 
            <apply> 
              <times/> 
              <cn>2</cn> 
              <ci>b</ci> 
            </apply> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
limit as x goes to a of sin(x + y)2b using &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <tendsto type='above'/> 
              <ci>x</ci> 
              <ci>a</ci> 
            </apply> 
          </condition> 
          <apply> 
            <times/> 
            <apply> 
              <sin/> 
              <apply> 
                <times/> 
                <ci>x</ci> 
                <ci>y</ci> 
              </apply> 
            </apply> 
            <apply> 
              <times/> 
              <cn>2</cn> 
              <ci>b</ci> 
            </apply> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
quotient 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <quotient/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
moment 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <moment/> 
          <degree> 
            <cn>3</cn> 
          </degree> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
selector 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <selector/> 
          <matrix> 
            <matrixrow> 
              <cn>1</cn> 
              <cn>2</cn> 
            </matrixrow> 
            <matrixrow> 
              <cn>3</cn> 
              <cn>4</cn> 
            </matrixrow> 
          </matrix> 
          <cn>1</cn> 
        </apply> 
      </math> 
    </p> 
    <p>
factorial 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <factorial/> 
          <ci>n</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
(a + b)! 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <factorial/> 
          <apply> 
            <plus/> 
            <apply> 
              <plus/> 
              <ci>n</ci> 
              <ci>m</ci> 
            </apply> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
inverse function 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <inverse/> 
          <ci>f</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
inverse matrix 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <apply> 
            <inverse/> 
            <ci type='matrix'>a</ci> 
          </apply> 
          <ci>A</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
conjugate 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <mrow> 
          <apply> 
            <conjugate/> 
            <apply> 
              <plus/> 
              <ci>x</ci> 
              <apply> 
                <times/> 
                <cn>&ImaginaryI;</cn> 
                <ci>y</ci> 
              </apply> 
            </apply> 
          </apply> 
        </mrow> 
      </math> 
    </p> 
    <p>
a + b + c 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <ci>a</ci> 
          <ci>b</ci> 
          <ci>c</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
integral (a + x)dx 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <int/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <lowlimit> 
            <cn>0</cn> 
          </lowlimit> 
          <uplimit> 
            <ci>a</ci> 
          </uplimit> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
-1 + 7 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <apply> 
            <minus/> 
            <cn>1</cn> 
          </apply> 
          <cn>7</cn> 
        </apply> 
      </math> 
    </p> 
    <p>
7 + (-1) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <cn>7</cn> 
          <apply> 
            <minus/> 
            <cn>1</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
max 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <max/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <and/> 
              <reln> 
                <gt/> 
                <ci>x</ci> 
                <cn>0</cn> 
              </reln> 
              <reln> 
                <lt/> 
                <ci>x</ci> 
                <cn>1</cn> 
              </reln> 
            </apply> 
          </condition> 
          <apply> 
            <minus/> 
            <ci>x</ci> 
            <apply> 
              <sin/> 
              <ci>x</ci> 
            </apply> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
lambda sin(x + 1) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <lambda> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <apply> 
            <sin/> 
            <apply> 
              <plus/> 
              <ci>x</ci> 
              <cn>1</cn> 
            </apply> 
          </apply> 
        </lambda> 
      </math> 
    </p> 
    <p>
lambda integral f(x)dx 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <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> 
      </math> 
    </p> 
    <p>
compose f and g 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <compose/> 
          <fn> 
            <ci>f</ci> 
          </fn> 
          <fn> 
            <ci>g</ci> 
          </fn> 
        </apply> 
      </math> 
    </p> 
    <p>
compose f and g (x) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <apply> 
            <compose/> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <fn> 
              <ci>g</ci> 
            </fn> 
          </apply> 
          <ci>x</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
f(g(x)) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <fn> 
            <ci>f</ci> 
          </fn> 
          <apply> 
            <fn> 
              <ci>g</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
composition of f and inverse of f eq identity using &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <eq/> 
          <apply> 
            <compose/> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <apply> 
              <inverse/> 
              <fn> 
                <ci>f</ci> 
              </fn> 
            </apply> 
          </apply> 
          <ident/> 
        </reln> 
      </math> 
    </p> 
    <p>
composition of f and inverse of f eq identity using &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <eq/> 
          <apply> 
            <compose/> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <apply> 
              <inverse/> 
              <fn> 
                <ci>f</ci> 
              </fn> 
            </apply> 
          </apply> 
          <ident/> 
        </apply> 
      </math> 
    </p> 
    <p>
e^x 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <exp/> 
          <ci>x</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
min(x, x not in B, x^2) using &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <min/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <notin/> 
              <ci>x</ci> 
              <ci type='set'>B</ci> 
            </reln> 
          </condition> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
min(x, x not in B, x^2) using &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <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> 
      </math> 
    </p> 
    <p>
a mod (b) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <rem/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
ab 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <times/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
gcd(a b, c) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <gcd/> 
          <ci>a</ci> 
          <ci>b</ci> 
          <ci>c</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
integral 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <int/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <lowlimit> 
            <cn>0</cn> 
          </lowlimit> 
          <uplimit> 
            <ci>a</ci> 
          </uplimit> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
abs(x) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <abs/> 
          <ci>x</ci> 
        </apply> 
      </math> 
    </p> 
    <p> tall abs(x) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <abs/> 
          <apply> 
            <divide/> 
            <ci>H</ci> 
            <ci>K</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
abs(x + y + z) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <abs/> 
          <apply> 
            <plus/> 
            <ci>x</ci> 
            <ci>y</ci> 
            <ci>z</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
x &gt; 0 and z &lt; 1 as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <and/> 
          <reln> 
            <gt/> 
            <ci>x</ci> 
            <cn>0</cn> 
          </reln> 
          <reln> 
            <lt/> 
            <ci>x</ci> 
            <cn>1</cn> 
          </reln> 
        </apply> 
      </math> 
    </p> 
    <p>
x &gt; 0 and z &lt; 1 as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <and/> 
          <apply> 
            <gt/> 
            <ci>x</ci> 
            <cn>0</cn> 
          </apply> 
          <apply> 
            <lt/> 
            <ci>x</ci> 
            <cn>1</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
a and b 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <and/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a or b 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <or/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a xor b 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <xor/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a eq b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <eq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a eq b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <eq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a neq b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <neq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a neq b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <neq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a &gt; b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <gt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a &gt; b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <gt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a &lt; b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <lt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a &lt; b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <lt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a &lt;= b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <geq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a &gt;= b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <geq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a &lt;= b with &lt;reln&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <leq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a &lt;= b with &lt;apply&gt; tag 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <leq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
set: {b, a, c} 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <set> 
          <ci>b</ci> 
          <ci>a</ci> 
          <ci>c</ci> 
        </set> 
      </math> 
    </p> 
    <p>
set with condition 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <set> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <lt/> 
              <ci>x</ci> 
              <cn>5</cn> 
            </reln> 
          </condition> 
        </set> 
      </math> 
    </p> 
    <p>
list: {b, a, c} 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <list> 
          <ci>b</ci> 
          <ci>a</ci> 
          <ci>c</ci> 
        </list> 
      </math> 
    </p> 
    <p>
list: {x|x &lt; 5} 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <list order='numeric'> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <lt/> 
              <ci>x</ci> 
              <cn>5</cn> 
            </reln> 
          </condition> 
        </list> 
      </math> 
    </p> 
    <p>
A union B 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <union/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
A intersect B 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <intersect/> 
          <ci type='set'>A</ci> 
          <ci type='set'>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
A intersect (B union C) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <intersect/> 
          <ci type='set'>A</ci> 
          <apply> 
            <union/> 
            <ci type='set'>B</ci> 
            <ci type='set'>C</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>integral
x in R as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <in/> 
          <ci>x</ci> 
          <ci type='set'>R</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
x in R as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <in/> 
          <ci>x</ci> 
          <ci type='set'>R</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
a in A 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <in/> 
          <ci>a</ci> 
          <ci type='set'>A</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a not in A as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notin/> 
          <ci>a</ci> 
          <ci>A</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
a not in A as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notin/> 
          <ci>a</ci> 
          <ci>A</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
not a 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <not/> 
          <ci>a</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
not (a and b) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <not/> 
          <apply> 
            <and/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
A -&gt; B (reln) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <implies/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
A -&gt; B (apply) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <implies/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
forall 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <forall/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <reln> 
            <eq/> 
            <apply> 
              <minus/> 
              <ci>x</ci> 
              <ci>x</ci> 
            </apply> 
            <cn>0</cn> 
          </reln> 
        </apply> 
      </math> 
    </p> 
    <p>
forall/and/lt/power 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <forall/> 
          <bvar> 
            <ci>p</ci> 
          </bvar> 
          <bvar> 
            <ci>q</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <and/> 
              <reln> 
                <in/> 
                <ci>p</ci> 
                <ci type='set'>Q</ci> 
              </reln> 
              <reln> 
                <in/> 
                <ci>q</ci> 
                <ci type='set'>Q</ci> 
              </reln> 
              <reln> 
                <lt/> 
                <ci>p</ci> 
                <ci>q</ci> 
              </reln> 
            </apply> 
          </condition> 
          <reln> 
            <lt/> 
            <ci>p</ci> 
            <apply> 
              <power/> 
              <ci>q</ci> 
              <cn>2</cn> 
            </apply> 
          </reln> 
        </apply> 
      </math> 
    </p> 
    <p>
forall/exists/and/plus 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <forall/> 
          <bvar> 
            <ci>n</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <and/> 
              <reln> 
                <in/> 
                <ci>n</ci> 
                <ci type='set'>Z</ci> 
              </reln> 
              <reln> 
                <gt/> 
                <ci>n</ci> 
                <cn>0</cn> 
              </reln> 
            </apply> 
          </condition> 
          <apply> 
            <exists/> 
            <bvar> 
              <ci>x</ci> 
            </bvar> 
            <bvar> 
              <ci>y</ci> 
            </bvar> 
            <bvar> 
              <ci>z</ci> 
            </bvar> 
            <condition> 
              <apply> 
                <and/> 
                <reln> 
                  <in/> 
                  <ci>x</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
                <reln> 
                  <in/> 
                  <ci>y</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
                <reln> 
                  <in/> 
                  <ci>z</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
              </apply> 
            </condition> 
            <reln> 
              <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> 
            </reln> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
exists 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <exists/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <reln> 
            <eq/> 
            <apply> 
              <fn> 
                <ci>f</ci> 
              </fn> 
              <ci>x</ci> 
            </apply> 
            <cn>0</cn> 
          </reln> 
        </apply> 
      </math> 
    </p> 
    <p>
forall/exists/and/plus 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <forall/> 
          <bvar> 
            <ci>n</ci> 
          </bvar> 
          <condition> 
            <apply> 
              <and/> 
              <reln> 
                <in/> 
                <ci>n</ci> 
                <ci type='set'>Z</ci> 
              </reln> 
              <reln> 
                <gt/> 
                <ci>n</ci> 
                <cn>0</cn> 
              </reln> 
            </apply> 
          </condition> 
          <apply> 
            <exists/> 
            <bvar> 
              <ci>x</ci> 
            </bvar> 
            <bvar> 
              <ci>y</ci> 
            </bvar> 
            <bvar> 
              <ci>z</ci> 
            </bvar> 
            <condition> 
              <apply> 
                <and/> 
                <reln> 
                  <in/> 
                  <ci>x</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
                <reln> 
                  <in/> 
                  <ci>y</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
                <reln> 
                  <in/> 
                  <ci>z</ci> 
                  <ci type='set'>Z</ci> 
                </reln> 
              </apply> 
            </condition> 
            <reln> 
              <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> 
            </reln> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
ln a 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <ln/> 
          <ci>a</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
log base 3 of x 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <log/> 
          <logbase> 
            <cn>3</cn> 
          </logbase> 
          <ci>x</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
integer 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <int/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <in/> 
              <ci>x</ci> 
              <ci type='set'>D</ci> 
            </reln> 
          </condition> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
diff 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <diff/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
partialdiff 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <partialdiff/> 
          <bvar> 
            <ci>x</ci> 
            <degree> 
              <cn>2</cn> 
            </degree> 
          </bvar> 
          <bvar> 
            <ci>y</ci> 
          </bvar> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
            <ci>y</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
integral 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <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> 
      </math> 
    </p> 
    <p>
partialdiff 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <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> 
      </math> 
    </p> 
    <p>
divide 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <divide/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
divide/plus/minus 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <divide/> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
          <apply> 
            <minus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
divide/plus/divide 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <divide/> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
          <apply> 
            <divide/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
A is subset of B as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <subset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
A is subset of B as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <subset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
A is proper subset of B as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <prsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
A is proper subset of B as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <prsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
A is not subset of B as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
A is not subset of B as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
A is not proper subset of B as &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notprsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln> 
      </math> 
    </p> 
    <p>
A is not proper subset of B as &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notprsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
Set difference 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <setdiff/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
Log base 3 of x + y 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <log/> 
          <logbase> 
            <cn>3</cn> 
          </logbase> 
          <apply> 
            <plus/> 
            <ci>x</ci> 
            <ci>y</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
Sum as x goes from a to b of f(x) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sum/> 
          <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> 
      </math> 
    </p> 
    <p>
sum 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sum/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <in/> 
              <ci>x</ci> 
              <ci type='set'>B</ci> 
            </reln> 
          </condition> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
product 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <product/> 
          <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> 
      </math> 
    </p> 
    <p>
product 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <product/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <condition> 
            <reln> 
              <in/> 
              <ci>x</ci> 
              <ci type='set'>B</ci> 
            </reln> 
          </condition> 
          <apply> 
            <fn> 
              <ci>f</ci> 
            </fn> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
tendsto with &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <tendsto type='above'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </reln> 
      </math> 
    </p> 
    <p>
tendsto with &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <tendsto type='above'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
tendsto with &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <tendsto/> 
          <vector> 
            <ci>x</ci> 
            <ci>y</ci> 
          </vector> 
          <vector> 
            <apply> 
              <fn> 
                <ci>f</ci> 
              </fn> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
            <apply> 
              <fn> 
                <ci>g</ci> 
              </fn> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
          </vector> 
        </reln> 
      </math> 
    </p> 
    <p>
tendsto with &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <tendsto/> 
          <vector> 
            <ci>x</ci> 
            <ci>y</ci> 
          </vector> 
          <vector> 
            <apply> 
              <fn> 
                <ci>f</ci> 
              </fn> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
            <apply> 
              <fn> 
                <ci>g</ci> 
              </fn> 
              <ci>x</ci> 
              <ci>y</ci> 
            </apply> 
          </vector> 
        </apply> 
      </math> 
    </p> 
    <p>
mean(X) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <mean/> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
root(a + b) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <root/> 
          <degree> 
            <ci>n</ci> 
          </degree> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
standard deviation 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sdev/> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
variance(X) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <variance/> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
median(X) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <median/> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
mode(X) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <mode/> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
degree 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <moment/> 
          <degree> 
            <cn>3</cn> 
          </degree> 
          <ci>X</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
vector 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <vector> 
          <cn>1</cn> 
          <cn>2</cn> 
          <cn>3</cn> 
          <ci>x</ci> 
        </vector> 
      </math> 
    </p> 
    <p>
matrix 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <matrix> 
          <matrixrow> 
            <cn>0</cn> 
            <cn>1</cn> 
            <cn>0</cn> 
            <cn>0</cn> 
          </matrixrow> 
          <matrixrow> 
            <cn>0</cn> 
            <cn>0</cn> 
            <cn>1</cn> 
            <cn>0</cn> 
          </matrixrow> 
          <matrixrow> 
            <cn>1</cn> 
            <cn>0</cn> 
            <cn>0</cn> 
            <cn>0</cn> 
          </matrixrow> 
        </matrix> 
      </math> 
    </p> 
    <p>
determinant 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <determinant/> 
          <ci type='matrix'>A</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
transpose 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <transpose/> 
          <ci type='matrix'>A</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
semantics 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <semantics> 
          <apply> 
            <plus/> 
            <apply> 
              <sin/> 
              <ci>x</ci> 
            </apply> 
            <cn>5</cn> 
          </apply> 
          <annotation encoding='TeX'>\sin x + 5 
          </annotation> 
        </semantics> 
      </math> 
    </p> 
    <p>
limit 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <limit/> 
          <bvar> 
            <ci>x</ci> 
          </bvar> 
          <lowlimit> 
            <cn>0</cn> 
          </lowlimit> 
          <apply> 
            <sin/> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
symbol check 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <cn type='real'>4.56</cn> 
          <cn type='integer'>4.56</cn> 
          <cn type='rational'>4 
            <sep/>5 
          </cn> 
          <cn type='complex-cartesian'>4 
            <sep/>5 
          </cn> 
          <cn type='complex-polar'>4.56 
            <sep/>4.56 
          </cn> 
          <cn type='constant'>&pi;</cn> 
          <cn>&ExponentialE;</cn> 
          <cn>&ee;</cn> 
          <cn>&ImaginaryI;</cn> 
          <cn>&ii;</cn> 
          <cn>&gamma;</cn> 
          <cn>&infin;</cn> 
          <cn>&infin;</cn> 
        </apply> 
      </math> 
    </p> 
    <p>
multiset 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <set type='multiset'> 
          <cn type='real'>4.56</cn> 
          <cn type='integer'>4.56</cn> 
          <cn type='rational'>4 
            <sep/>5 
          </cn> 
          <cn type='complex-cartesian'>4 
            <sep/>5 
          </cn> 
          <cn type='complex-polar'>4.56 
            <sep/>4.56 
          </cn> 
          <cn type='constant'>&pi;</cn> 
          <cn>&ExponentialE;</cn> 
          <cn>&ee;</cn> 
          <cn>&ImaginaryI;</cn> 
          <cn>&ii;</cn> 
          <cn>&gamma;</cn> 
          <cn>&infin;</cn> 
          <cn>&infin;</cn> 
        </set> 
      </math> 
    </p> 
    <p>
tendsto type = "above" with &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <tendsto type='above'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </reln> 
      </math> 
    </p> 
    <p>
tendsto type = "above" with &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <tendsto type='above'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
tendsto type = "below" with &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <tendsto type='below'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </reln> 
      </math> 
    </p> 
    <p>
tendsto type = "below" with &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <tendsto type='below'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
tendsto type = "two-sided" with &lt;reln&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <tendsto type='two-sided'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </reln> 
      </math> 
    </p> 
    <p>
tendsto type = "two-sided" with &lt;apply&gt; 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <tendsto type='two-sided'/> 
          <apply> 
            <power/> 
            <ci>x</ci> 
            <cn>2</cn> 
          </apply> 
          <apply> 
            <power/> 
            <ci>a</ci> 
            <cn>2</cn> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
type check 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <ci type='real'>x</ci> 
          <ci type='integer'>x</ci> 
          <ci type='rational'>x</ci> 
          <ci type='complex-cartesian'>y</ci> 
          <ci type='complex-polar'>&theta;</ci> 
          <ci type='vector'>v</ci> 
          <ci type='constant'>&pi;</ci> 
          <ci>&ExponentialE;</ci> 
          <ci>&ee;</ci> 
          <ci>&ImaginaryI;</ci> 
          <ci>&ii;</ci> 
          <ci>&gamma;</ci> 
          <ci>&infin;</ci> 
          <ci>&infin;</ci> 
        </apply> 
      </math> 
    </p> 
    <p>
sin + cos 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <apply> 
            <sin/> 
            <ci>x</ci> 
          </apply> 
          <apply> 
            <cos/> 
            <ci>x</ci> 
          </apply> 
        </apply> 
      </math> 
    </p> 
    <p>
f(x) 
      <br /> 
      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <fn> 
            <ci>f</ci> 
          </fn> 
          <ci>x</ci> 
        </apply> 
      </math> 
    </p>