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, not, or, true, xor

Date:

2004-03-30

Version:

3

Review Date:

2006-03-30

Status:

official

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     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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     holder permits you to develop any derived work from this document
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          this document, and maintains a prominent reference in the 
          work to the original source.
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          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
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          Dictionary Group whose name is, for example, `math' containing
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     If you have questions about this license please contact the OpenMath
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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):

xml prefix mathml

<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>

equivalent (equivalent ( A, B) , and (implies ( A, B) , implies ( B, A) ) )

$$A\equiv B\equiv ((A\Rightarrow B)\wedge (B\Rightarrow A))$$

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):

xml prefix mathml

<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>

forall [ x ] . (eq (not (not ( x) ) , x) )

$$\forall \hspace{0.17em}x.\neg \neg x=x$$

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):

xml prefix mathml

<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>

forall [ x ] . (eq (and ( x, not ( x) ) , false) )

$$\forall \hspace{0.17em}x.x\wedge \neg x=F$$

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):

xml prefix mathml

<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>

forall [ x ] . (eq (xor ( x, x) , false) )

$$\forall \hspace{0.17em}x.xxorx=F$$

Commented Mathematical property (CMP):

for all x | x xor not(x) = true

Formal Mathematical property (FMP):

xml prefix mathml

<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>

forall [ x ] . (eq (xor ( x, not ( x) ) , true) )

$$\forall \hspace{0.17em}x.xxor\neg x=T$$

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):

xml prefix mathml

<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>

forall [ x ] . (eq (or ( x, not ( x) ) , true) )

$$\forall \hspace{0.17em}x.x\vee \neg x=T$$

Commented Mathematical property (CMP):

for all a,b | not(a and b)= (not(a) or not(b))

Formal Mathematical property (FMP):

xml prefix mathml

<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>

forall [ a b ] . (eq (not (and ( a, b) ) , or (not ( a) , not ( b) ) ) )

$$\forall \hspace{0.17em}a,b.\neg (a\wedge b)=\neg a\vee \neg b$$

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):

xml prefix mathml

<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>

forall [ x ] . (implies (false, x) )

$$\forall \hspace{0.17em}x.F\Rightarrow x$$

Signatures:

sts

true

Role:

constant

Description:

This symbol represents the boolean value true.

Commented Mathematical property (CMP):

not true = false

Formal Mathematical property (FMP):

xml prefix mathml

<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>

eq (not (true) , false)

$$\neg T=F$$

Signatures:

sts

false

Role:

constant

Description:

This symbol represents the boolean value false.

Commented Mathematical property (CMP):

not false = true

Formal Mathematical property (FMP):

xml prefix mathml

<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>

eq (not (false) , true)

$$\neg F=T$$

Signatures:

sts