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logic1

OpenMath Content Dictionary: logic1
Canonical URL:
http://www.openmath.org/cd/logic1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
logic1.ocd
CD as XML Encoded OpenMath:
logic1.omcd
Defines:
and, equivalent, false, implies, nand, nor, not, or, true, xnor, xor
Date:
2004-03-30
Version:
4
Review Date:
2006-03-30
Status:
official 
     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

This CD holds the basic logic functions.

equivalent
Role:
application
Description:

This symbol is used to show that two boolean expressions are logically equivalent, that is have the same boolean value for any inputs.

Commented Mathematical property (CMP):
The condition (A is equivalent to B) is equivalent to the condition that (A implies B and B implies A)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="logic1" name="equivalent"/>
      <OMV name="A"/>
      <OMV name="B"/>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="logic1" name="implies"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
      <OMA>
        <OMS cd="logic1" name="implies"/>
	<OMV name="B"/>
	<OMV name="A"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="logic1">equivalent</csymbol><ci>A</ci><ci>B</ci></apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="logic1">implies</csymbol><ci>A</ci><ci>B</ci></apply>
   <apply><csymbol cd="logic1">implies</csymbol><ci>B</ci><ci>A</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(logic1.equivalent($A, $B), ($A ==> $B) and ($B ==> $A))

A ≡ B ≡ ( ( A ⇒ B ) ∧ ( B ⇒ A ) )

Signatures:
sts
not
Role:
application
Description:

This symbol represents the logical not function which takes one boolean argument, and returns the opposite boolean value.

Commented Mathematical property (CMP):
for all x | not(not(x))=x
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="logic1" name="not"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">not</csymbol>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <ci>x</ci>
  </apply>
 </bind>
</math>

quant1.forall[$x -> not( not($x)) = $x]

Signatures:
sts
and
Role:
application
Description:

This symbol represents the logical and function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if all arguments are true or false otherwise.

Commented Mathematical property (CMP):
for all x | x and not(x) = false
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="and"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="false"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">false</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> ($x and not($x)) = logic1.false]

Signatures:
sts
nand
Role:
application
Description:

This symbol represents the logical nand function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if all arguments are true or true otherwise.

Commented Mathematical property (CMP):
for all x | x nand not(x) = true
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="nand"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">nand</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">true</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.nand($x, not($x)) = logic1.true]

∀   x . nand ⁡ ( x , ¬ x ) = T

Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
    <OMV name="y"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="nand"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="logic1" name="not"/>
      <OMA><OMS cd="logic1" name="and"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">nand</csymbol><ci>x</ci><ci>y</ci></apply>
   <apply><csymbol cd="logic1">not</csymbol>
    <apply><csymbol cd="logic1">and</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> logic1.nand($x, $y) = not($x and $y)]

∀   x , y . nand ⁡ ( x , y ) = ¬ ( x ∧ y )

Signatures:
sts
xor
Role:
application
Description:

This symbol represents the logical xor function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if there are an odd number of true arguments or false otherwise.

Commented Mathematical property (CMP):
for all x | x xor x = false
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="xor"/>
      <OMV name="x"/>
      <OMV name="x"/>
    </OMA>
    <OMS cd="logic1" name="false"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">xor</csymbol><ci>x</ci><ci>x</ci></apply>
   <csymbol cd="logic1">false</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.xor($x, $x) = logic1.false]

Commented Mathematical property (CMP):
for all x | x xor not(x) = true
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="xor"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">xor</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">true</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.xor($x, not($x)) = logic1.true]

Signatures:
sts
xnor
Role:
application
Description:

This symbol represents the logical xnor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if there are an odd number of true arguments or true otherwise.

Commented Mathematical property (CMP):
for all x | x xnor x = true
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="xnor"/>
      <OMV name="x"/>
      <OMV name="x"/>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">xnor</csymbol><ci>x</ci><ci>x</ci></apply>
   <csymbol cd="logic1">true</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.xnor($x, $x) = logic1.true]

∀   x . xnor ⁡ ( x , x ) = T

Commented Mathematical property (CMP):
for all x | x xnor not(x) = false
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="xnor"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="false"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">xnor</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">false</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.xnor($x, not($x)) = logic1.false]

∀   x . xnor ⁡ ( x , ¬ x ) = F

Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
    <OMV name="y"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="xnor"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="logic1" name="not"/>
      <OMA><OMS cd="logic1" name="xor"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">xnor</csymbol><ci>x</ci><ci>y</ci></apply>
   <apply><csymbol cd="logic1">not</csymbol>
    <apply><csymbol cd="logic1">xor</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> logic1.xnor($x, $y) = not(logic1.xor($x, $y))]

∀   x , y . xnor ⁡ ( x , y ) = ¬ ( x xor y )

Signatures:
sts
or
Role:
application
Description:

This symbol represents the logical or function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if any of the arguments are true or false otherwise.

Commented Mathematical property (CMP):
for all x | x or not(x) = true
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="or"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">or</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">true</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> ($x > not($x)) = logic1.true]

Commented Mathematical property (CMP):
for all a,b | not(a and b)= (not(a) or not(b))
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="a"/>
    <OMV name="b"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="logic1" name="not"/>
      <OMA>
        <OMS cd="logic1" name="and"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="or"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">not</csymbol>
    <apply><csymbol cd="logic1">and</csymbol><ci>a</ci><ci>b</ci></apply>
   </apply>
   <apply><csymbol cd="logic1">or</csymbol>
    <apply><csymbol cd="logic1">not</csymbol><ci>a</ci></apply>
    <apply><csymbol cd="logic1">not</csymbol><ci>b</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$a, $b -> not($a and $b) = ( not($a) > not($b))]

∀   a , b . ¬ ( a ∧ b ) = ¬ a ∨ ¬ b

Signatures:
sts
nor
Role:
application
Description:

This symbol represents the logical nor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if any of the arguments are true or true otherwise.

Commented Mathematical property (CMP):
for all x | x nor not(x) = false
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMBIND><OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="nor"/>
      <OMV name="x"/>
      <OMA><OMS cd="logic1" name="not"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="false"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">nor</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="logic1">not</csymbol><ci>x</ci></apply>
   </apply>
   <csymbol cd="logic1">false</csymbol>
  </apply>
 </bind>
</math>

quant1.forall[$x -> logic1.nor($x, not($x)) = logic1.false]

∀   x . nor ⁡ ( x , ¬ x ) = F

Commented Mathematical property (CMP):
for all a,b | a and b = (not(a) nor not(b))
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="a"/>
    <OMV name="b"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="nor"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">and</csymbol><ci>a</ci><ci>b</ci></apply>
   <apply><csymbol cd="logic1">nor</csymbol>
    <apply><csymbol cd="logic1">not</csymbol><ci>a</ci></apply>
    <apply><csymbol cd="logic1">not</csymbol><ci>b</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$a, $b -> ($a and $b) = logic1.nor( not($a), not($b))]

∀   a , b . a ∧ b = nor ⁡ ( ¬ a , ¬ b )

Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
    <OMV name="y"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="logic1" name="nor"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="logic1" name="not"/>
      <OMA><OMS cd="logic1" name="or"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="logic1">nor</csymbol><ci>x</ci><ci>y</ci></apply>
   <apply><csymbol cd="logic1">not</csymbol>
    <apply><csymbol cd="logic1">or</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> logic1.nor($x, $y) = not($x > $y)]

∀   x , y . nor ⁡ ( x , y ) = ¬ ( x ∨ y )

Signatures:
sts
implies
Role:
application
Description:

This symbol represents the logical implies function which takes two boolean expressions as arguments. It evaluates to false if the first argument is true and the second argument is false, otherwise it evaluates to true.

Commented Mathematical property (CMP):
for all x | false implies x
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMS cd="logic1" name="false"/>
    <OMV name="x"/>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol><csymbol cd="logic1">false</csymbol><ci>x</ci></apply>
 </bind>
</math>

quant1.forall[$x -> logic1.false ==> $x]

Signatures:
sts
true
Role:
constant
Description:

This symbol represents the boolean value true.

Commented Mathematical property (CMP):
not true = false
Formal Mathematical property (FMP): 
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
	<OMS cd="logic1" name="true"/>
      </OMA>
      <OMS cd="logic1" name="false"/>
    </OMA>
  </OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="logic1">not</csymbol><csymbol cd="logic1">true</csymbol></apply>
  <csymbol cd="logic1">false</csymbol>
 </apply>
</math>

not(logic1.true) = logic1.false

Signatures:
sts
false
Role:
constant
Description:

This symbol represents the boolean value false.

Commented Mathematical property (CMP):
not false = true
Formal Mathematical property (FMP): 
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
	<OMS cd="logic1" name="false"/>
      </OMA>
      <OMS cd="logic1" name="true"/>
    </OMA>
  </OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="logic1">not</csymbol><csymbol cd="logic1">false</csymbol></apply>
  <csymbol cd="logic1">true</csymbol>
 </apply>
</math>

not(logic1.false) = logic1.true

Signatures:
sts

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