OpenMath Content Dictionary: relation1

Canonical URL:
http://www.openmath.org/cd/relation1.ocd
CD File:
relation1.ocd
CD as XML Encoded OpenMath:
relation1.omcd
Defines:
approx, eq, geq, gt, leq, lt, neq
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
arith1, logic1, nums1

This CD holds the common arithmetic relations. It is intended to be `compatible' with the appropriate MathML elements.


eq

This symbol represents the binary equality function.

Commented Mathematical property (CMP):
a=b and b=c implies a=c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMV name="a"/>
    <OMV name="c"/>
  </OMA>
</OMA>
</OMOBJ>

implies (and (eq ( a, b) , eq ( b, c) ) , eq ( a, c) )

Example:
An example which represents the statement 1 + 2 = 3.
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 3 </OMI>
  </OMA>
</OMOBJ>

eq (plus ( 1 , 2 ) , 3 )

Signatures:
sts


[Next: lt] [Last: approx] [Top]

lt

This symbol represents the binary less than function which returns true if the first argument is less than the second, it returns false otherwise.

Commented Mathematical property (CMP):
a<b and b<c implies a<c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="lt"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="lt"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="lt"/>
    <OMV name="a"/>
    <OMV name="c"/>
  </OMA>
</OMA>
</OMOBJ>

implies (and (lt ( a, b) , lt ( b, c) ) , lt ( a, c) )

Example:
An example which represents the statement 1 + 2 < 4
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="lt"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 4 </OMI>
  </OMA>
</OMOBJ>

lt (plus ( 1 , 2 ) , 4 )

Signatures:
sts


[Next: gt] [Previous: eq] [Top]

gt

This symbol represents the binary greater than function which returns true if the first argument is greater than the second, it returns false otherwise.

Commented Mathematical property (CMP):
a>b and b>c implies a>c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="gt"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="gt"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="gt"/>
    <OMV name="a"/>
    <OMV name="c"/>
  </OMA>
</OMA>
</OMOBJ>

implies (and (gt ( a, b) , gt ( b, c) ) , gt ( a, c) )

Example:
An example which represents the statement 1 + 2 > 2
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="gt"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 2 </OMI>
  </OMA>
</OMOBJ>

gt (plus ( 1 , 2 ) , 2 )

Signatures:
sts


[Next: neq] [Previous: lt] [Top]

neq

This symbol represents the binary inequality function.

Commented Mathematical property (CMP):
it is not true that a=/b and b=/c implies a=/c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="not"/>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="relation1" name="neq"/>
	<OMV name="a"/>
	<OMV name="b"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="neq"/>
	<OMV name="b"/>
	<OMV name="c"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="neq"/>
      <OMV name="a"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

not (implies (and (neq ( a, b) , neq ( b, c) ) , neq ( a, c) ) )

Example:
An example which represents the statement 1 + 2 not = 2
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="neq"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 2 </OMI>
  </OMA>
</OMOBJ>

neq (plus ( 1 , 2 ) , 2 )

Signatures:
sts


[Next: leq] [Previous: gt] [Top]

leq

This symbol represents the binary less than or equal to function which returns true if the first argument is less than or equal to the second, it returns false otherwise.

Commented Mathematical property (CMP):
a<=b and b<=c implies a<=c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="leq"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="leq"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="leq"/>
    <OMV name="a"/>
    <OMV name="c"/>
  </OMA>
</OMA>
</OMOBJ>

implies (and (leq ( a, b) , leq ( b, c) ) , leq ( a, c) )

Example:
An example which represents the statement 1 + 2 <= 4
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="leq"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 4 </OMI>
  </OMA>
</OMOBJ>

leq (plus ( 1 , 2 ) , 4 )

Signatures:
sts


[Next: geq] [Previous: neq] [Top]

geq

This symbol represents the binary greater than or equal to function which returns true if the first argument is greater than or equal to the second, it returns false otherwise.

Commented Mathematical property (CMP):
a>=b and b>=c implies a>=c
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="geq"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="geq"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="geq"/>
    <OMV name="a"/>
    <OMV name="c"/>
  </OMA>
</OMA>
</OMOBJ>

implies (and (geq ( a, b) , geq ( b, c) ) , geq ( a, c) )

Example:
An example which represents the statement 1 + 2 >= 3
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="geq"/>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
    <OMI> 3 </OMI>
  </OMA>
</OMOBJ>

geq (plus ( 1 , 2 ) , 3 )

Signatures:
sts


[Next: approx] [Previous: leq] [Top]

approx

This symbol is used to denote the approximate equality of its two arguments.

Example:
\pi is approximately 355/113
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="approx"/>
    <OMS cd="nums1" name="pi"/>
    <OMA>
      <OMS cd="nums1" name="rational"/>
      <OMI> 355 </OMI>
      <OMI> 113 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

approx (pi, rational ( 355 , 113 ) )

Signatures:
sts


[First: eq] [Previous: geq] [Top]