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<?xml-stylesheet type="text/xsl" href="../../../style/mathml.xsl"?>
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
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  Mathematics, Keio University). All Rights Reserved. See
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  <head>
    <title> MathML TestSuite, Version 2: complex4</title>
  </head>
  <body>
      <table width="100%" cellpadding="8">
        <tr>
          <td>
            <a href="../../TortureTests/Complexity/complex3.xml">
    prev </a>
          </td>
          <td style ="text-align: right">
            <a href="../../TortureTests/Complexity/simplePres.xml">
    next</a>
          </td>
        </tr>
      </table>
      <table style="background: #cccccc" cellpadding="4">
        <tr>
          <td style ="text-align: right" valign="top">File:</td>
          <td>/TortureTests/Complexity/complex4.xml</td>
        </tr>
        <tr>
          <td style ="text-align: right" valign="top">Author:</td>
          <td>Design Science, Inc. (D. Doyle, R. Miner)</td>
        </tr>
        <tr>
          <td style ="text-align: right" valign="top">Description:</td>
          <td>Test of most content elements in a single page.</td>
        </tr>
        <tr>
          <td style ="text-align: right" valign="top">Sample Rendering:</td>
          <td>
            N/A
          </td>
        </tr>
      </table>
      <p><b>Your browser's rendering</b>:</p>
  <p>  a + b - c:</p>
      
     <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>
    
    
x + yz + z:
      
</p>      <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>
    
    
x * (y + z) * z:
      
</p>      <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>
    
    
sin(a + b):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sin/> 
          <apply> 
            <plus/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply>
      </math> <p>
    
    
sin(x(y + z)z):
      
</p>      <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>
    
    
sin(xy)2b:
      
</p>      <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>
    
    
(x + y) ^ (n - 3):
      
</p>      <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>
    
    
limit as x goes to a of sin x using &#x003c;reln>:
      
</p>      <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>
    
    
limit as x goes to a of sin x using &#x003c;apply>:
      
</p>      <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>
    
    
limit as x goes to a of sin(x + y) using &#x003c;reln>:
      
</p>      <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>
    
    
limit as x goes to a of sin(x + y) using &#x003c;apply>:
      
</p>      <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>
    
    
limit as x goes to a of sin(x + y)2b using &#x003c;reln>:
      
</p>      <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>
    
    
limit as x goes to a of sin(x + y)2b using &#x003c;apply>:
      
</p>      <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>
    
    
quotient:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <quotient/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
moment:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <moment/> 
          <degree> 
            <cn>3</cn> 
          </degree> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
selector:
      
</p>      <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>
    
    
factorial:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <factorial/> 
          <ci>n</ci> 
        </apply>
      </math> <p>
    
    
(a + b)!:
      
</p>      <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>
    
    
inverse function:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <inverse/> 
          <ci>f</ci> 
        </apply>
      </math> <p>
    
    
inverse matrix:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <apply> 
            <inverse/> 
            <ci type="matrix">a</ci> 
          </apply> 
          <ci>A</ci> 
        </apply>
      </math> <p>
    
    
conjugate:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <mrow> 
          <apply> 
            <conjugate/> 
            <apply> 
              <plus/> 
              <ci>x</ci> 
              <apply> 
                <times/> 
                <cn>&#x02148;</cn> 
                <ci>y</ci> 
              </apply> 
            </apply> 
          </apply> 
        </mrow>
      </math> <p>
    
    
a + b + c:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <ci>a</ci> 
          <ci>b</ci> 
          <ci>c</ci> 
        </apply>
      </math> <p>
    
    
integral (a + x)dx:
      
</p>      <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>
    
    
-1 + 7:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <apply> 
            <minus/> 
            <cn>1</cn> 
          </apply> 
          <cn>7</cn> 
        </apply>
      </math> <p>
    
    
7 + (-1):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <plus/> 
          <cn>7</cn> 
          <apply> 
            <minus/> 
            <cn>1</cn> 
          </apply> 
        </apply>
      </math> <p>
    
    
max:
      
</p>      <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>
    
    
lambda sin(x + 1):
      
</p>      <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>
    
    
lambda integral f(x)dx:
      
</p>      <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>
    
    
compose f and g:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <compose/> 
          <fn> 
            <ci>f</ci> 
          </fn> 
          <fn> 
            <ci>g</ci> 
          </fn> 
        </apply>
      </math> <p>
    
    
compose f and g (x):
      
</p>      <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>
    
    
f(g(x)):
      
</p>      <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>
    
    
composition of f and inverse of f eq identity using &#x003c;reln>:
      
</p>      <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>
    
    
composition of f and inverse of f eq identity using &#x003c;apply>:
      
</p>      <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>
    
    
e^x:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <exp/> 
          <ci>x</ci> 
        </apply>
      </math> <p>
    
    
min(x, x not in B, x^2) using &#x003c;reln>:
      
</p>      <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>
    
    
min(x, x not in B, x^2) using &#x003c;apply>:
      
</p>      <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>
    
    
a mod (b):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <rem/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
ab:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <times/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
gcd(a b, c):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <gcd/> 
          <ci>a</ci> 
          <ci>b</ci> 
          <ci>c</ci> 
        </apply>
      </math> <p>
    
    
integral:
      
</p>      <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>
    
    
abs(x):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <abs/> 
          <ci>x</ci> 
        </apply>
      </math> <p>
    
     tall abs(x):

</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <abs/> 
          <apply> 
            <divide/> 
            <ci>H</ci> 
            <ci>K</ci> 
          </apply> 
        </apply>
      </math> <p>
    
    
abs(x + y + z):
      
</p>      <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>
    
    
x &#x0003E; 0 and z &lt; 1 as &lt;reln>:
      
</p>      <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>
    
    
x &#x0003E; 0 and z &lt; 1 as &lt;apply>:
      
</p>      <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>
    
    
a and b:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <and/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a or b:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <or/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a xor b:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <xor/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a eq b with &#x003c;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <eq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a eq b with &#x003c;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <eq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a neq b with &#x003c;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <neq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a neq b with &#x003c;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <neq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a &#x0003E; b with &lt;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <gt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a &#x0003E; b with &lt;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <gt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a &#x003c; b with &lt;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <lt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a &#x003c; b with &lt;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <lt/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a &#x003c;= b with &lt;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <geq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a &#x0003E;= b with &lt;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <geq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
a &#x003c;= b with &lt;reln> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <leq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </reln>
      </math> <p>
    
    
a &#x003c;= b with &lt;apply> tag:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <leq/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
set: {b, a, c}:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <set> 
          <ci>b</ci> 
          <ci>a</ci> 
          <ci>c</ci> 
        </set>
      </math> <p>
    
    
set with condition:
      
</p>      <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>
    
    
list: {b, a, c}:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <list> 
          <ci>b</ci> 
          <ci>a</ci> 
          <ci>c</ci> 
        </list>
      </math> <p>
    
    
list: {x|x &#x003c; 5}:
      
</p>      <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>
    
    
A union B:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <union/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
A intersect B:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <intersect/> 
          <ci type="set">A</ci> 
          <ci type="set">B</ci> 
        </apply>
      </math> <p>
    
    
A intersect (B union C):
      
</p>      <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>
    
    integral
x in R as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
          <reln> 
            <in/> 
            <ci>x</ci> 
            <ci type="set">R</ci> 
          </reln> 
      </math> <p>
    
    
x in R as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
          <apply> 
            <in/> 
            <ci>x</ci> 
            <ci type="set">R</ci> 
          </apply> 
      </math> <p>
    
a in A:

</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <in/> 
          <ci>a</ci> 
          <ci type="set">A</ci> 
        </reln>
      </math> <p>
    
    
a not in A as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notin/> 
          <ci>a</ci> 
          <ci>A</ci> 
        </reln>
      </math> <p>
    
    
a not in A as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notin/> 
          <ci>a</ci> 
          <ci>A</ci> 
        </apply>
      </math> <p>
    
    
not a:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <not/> 
          <ci>a</ci> 
        </apply>
      </math> <p>
    
    
not (a and b):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <not/> 
          <apply> 
            <and/> 
            <ci>a</ci> 
            <ci>b</ci> 
          </apply> 
        </apply>
      </math> <p>
    
    
A -&#x0003E; B (reln):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <implies/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln>
      </math> <p>
    
    
A -&#x0003E; B (apply):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <implies/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
forall:
      
</p>      <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>
    
    
forall/and/lt/power:
      
</p>      <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>
    
    
forall/exists/and/plus:
      
</p>      <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>
    
    
exists:
      
</p>      <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>
    
    
forall/exists/and/plus:
      
</p>      <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>
    
    
ln a:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <ln/> 
          <ci>a</ci> 
        </apply>
      </math> <p>
    
    
log base 3 of x:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <log/> 
          <logbase> 
            <cn>3</cn> 
          </logbase> 
          <ci>x</ci> 
        </apply>
      </math> <p>
    
    
integer:
      
</p>      <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>
    
    
diff:
      
</p>      <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>
    
    
partialdiff:
      
</p>      <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>
    
    
integral:
      
</p>      <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>
    
    
partialdiff:
      
</p>      <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>
    
    
divide:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <divide/> 
          <ci>a</ci> 
          <ci>b</ci> 
        </apply>
      </math> <p>
    
    
divide/plus/minus:
      
</p>      <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>
    
    
divide/plus/divide:
      
</p>      <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>
    
    
A is subset of B as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <subset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln>
      </math> <p>
    
    
A is subset of B as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <subset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
A is proper subset of B as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <prsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln>
      </math> <p>
    
    
A is proper subset of B as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <prsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
A is not subset of B as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln>
      </math> <p>
    
    
A is not subset of B as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
A is not proper subset of B as &#x003c;reln>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <reln> 
          <notprsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </reln>
      </math> <p>
    
    
A is not proper subset of B as &#x003c;apply>:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <notprsubset/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
Set difference:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <setdiff/> 
          <ci>A</ci> 
          <ci>B</ci> 
        </apply>
      </math> <p>
    
    
Log base 3 of x + y:
      
</p>      <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>
    
    
Sum as x goes from a to b of f(x):
      
</p>      <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>
    
    
sum:
      
</p>      <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>
    
    
product:
      
</p>      <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>
    
    
product:
      
</p>      <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>
    
    
tendsto with &#x003c;reln>:
      
</p>      <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>
    
    
tendsto with &#x003c;apply>:
      
</p>      <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>
    
    
tendsto with &#x003c;reln>:
      
</p>      <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>
    
    
tendsto with &#x003c;apply>:
      
</p>      <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>
    
    
mean(X):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <mean/> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
root(a + b):
      
</p>      <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>
    
    
standard deviation:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <sdev/> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
variance(X):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <variance/> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
median(X):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <median/> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
mode(X):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <mode/> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
degree:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <moment/> 
          <degree> 
            <cn>3</cn> 
          </degree> 
          <ci>X</ci> 
        </apply>
      </math> <p>
    
    
vector:
      
</p>      <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>
    
    
matrix:
      
</p>      <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>
    
    
determinant:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <determinant/> 
          <ci type="matrix">A</ci> 
        </apply>
      </math> <p>
    
    
transpose:
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <transpose/> 
          <ci type="matrix">A</ci> 
        </apply>
      </math> <p>
    
    
semantics:
      
</p>      <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>
    
    
limit:
      
</p>      <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>
    
    
symbol check:
      
</p>      <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">&#x003C0;</cn> 
          <cn>&#x02147;</cn> 
          <cn>&#x02147;</cn> 
          <cn>&#x02148;</cn> 
          <cn>&#x02148;</cn> 
          <cn>&#x003B3;</cn> 
          <cn>&#x0221E;</cn> 
        </apply>
      </math> <p>
    
    
multiset:
      
</p>      <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">&#x003C0;</cn> 
          <cn>&#x02147;</cn> 
          <cn>&#x02147;</cn> 
          <cn>&#x02148;</cn> 
          <cn>&#x02148;</cn> 
          <cn>&#x003B3;</cn> 
          <cn>&#x0221E;</cn> 
        </set>
      </math> <p>
    
    
tendsto type = "above" with &#x003c;reln>:
      
</p>      <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>
    
    
tendsto type = "above" with &#x003c;apply>:
      
</p>      <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>
    
    
tendsto type = "below" with &#x003c;reln>:
      
</p>      <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>
    
    
tendsto type = "below" with &#x003c;apply>:
      
</p>      <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>
    
    
tendsto type = "two-sided" with &#x003c;reln>:
      
</p>      <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>
    
    
tendsto type = "two-sided" with &#x003c;apply>:
      
</p>      <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>
    
    
type check:
      
</p>      <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">&#x003B8;</ci> 
          <ci type="vector">v</ci> 
          <ci type="constant">&#x003C0;</ci> 
          <ci>&#x02147;</ci> 
          <ci>&#x02147;</ci> 
          <ci>&#x02148;</ci> 
          <ci>&#x02148;</ci> 
          <ci>&#x003B3;</ci> 
          <ci>&#x0221E;</ci> 
        </apply>
      </math> <p>
    
    
sin + cos:
      
</p>      <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>
    
    
f(x):
      
</p>      <math xmlns="http://www.w3.org/1998/Math/MathML"> 
        <apply> 
          <fn> 
            <ci>f</ci> 
          </fn> 
          <ci>x</ci> 
        </apply>
      </math> 
    
  <hr/>
    <p>
      <b>Source Code</b>: Omitted</p>
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