OpenMath Content Dictionary: logic1

Canonical URL:
http://www.openmath.org/cd/logic1.ocd
CD File:
logic1.ocd
CD as XML Encoded OpenMath:
logic1.omcd
Defines:
and, equivalent, false, implies, not, or, true, xor
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
quant1, relation1 

     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.

This CD holds the basic logic functions.

equivalent

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>
  <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) ) )

Signatures:
sts

[Next: not] [Last: false] [Top]

not

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>
<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) )

Signatures:
sts

[Next: and] [Previous: equivalent] [Top]

and

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>
<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) )

Signatures:
sts

[Next: xor] [Previous: not] [Top]

xor

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>
  <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) )

Commented Mathematical property (CMP):
for all x | x xor not(x) = true
Formal Mathematical property (FMP): 
<OMOBJ><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) )

Signatures:
sts

[Next: or] [Previous: and] [Top]

or

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><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) )

Commented Mathematical property (CMP):
for all a,b | not(a and b)= (not(a) or not(b))
Formal Mathematical property (FMP): 
<OMOBJ>
<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) ) ) )

Signatures:
sts

[Next: implies] [Previous: xor] [Top]

implies

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>
<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) )

Signatures:
sts

[Next: true] [Previous: or] [Top]

true

This symbol represents the boolean value true.

Commented Mathematical property (CMP):
not true = false
Formal Mathematical property (FMP): 
<OMOBJ>
    <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)

Signatures:
sts

[Next: false] [Previous: implies] [Top]

false

This symbol represents the boolean value false.

Commented Mathematical property (CMP):
not false = true
Formal Mathematical property (FMP): 
<OMOBJ>
    <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)

Signatures:
sts

[First: equivalent] [Previous: true] [Top]