Difference between revisions of "Exclusive disjunction"

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'''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
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The [[truth table]] of '''p XOR q''' (also written as '''p + q''' or '''p &ne; q''') is as follows:
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'''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true.
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The [[truth table]] of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q~\!</math> or <math>p \ne q,\!</math> appears below:
  
 
<br>
 
<br>
  
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
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|+ '''Exclusive Disjunction'''
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|+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math>
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|- style="background:#e6e6ff"
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|- style="height:40px; background:#f0f0ff"
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! style="width:15%" | p
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| style="width:33%" | <math>p\!</math>
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! style="width:15%" | q
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| style="width:33%" | <math>q\!</math>
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! style="width:15%" | p XOR q
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| style="width:33%" | <math>p ~\operatorname{XOR}~ q</math>
 
|-
 
|-
−
| F || F || F
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| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
 
|-
 
|-
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| F || T || T
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| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
 
|-
 
|-
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| T || F || T
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| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
 
|-
 
|-
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| T || T || F
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| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
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The following equivalents can then be deduced:
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The following equivalents may then be deduced:
  
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: <math>\begin{matrix}
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{| align="center" cellspacing="10" width="90%"
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p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
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|
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\\
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<math>\begin{matrix}
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       & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
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p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q)
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\\
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\\[6pt]
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       & = & (p \lor q) & \land & (\lnot p \lor \lnot q)
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\\[6pt]
 
       & = & (p \lor q) & \land & \lnot (p \land q)
 
       & = & (p \lor q) & \land & \lnot (p \land q)
 
\end{matrix}</math>
 
\end{matrix}</math>
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|}
  
 
==Syllabus==
 
==Syllabus==
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===Focal nodes===
 
===Focal nodes===
  
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* [[Inquiry Live]]
 
* [[Inquiry Live]]
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{{col-break}}
 
 
* [[Logic Live]]
 
* [[Logic Live]]
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{{col-end}}
 
  
 
===Peer nodes===
 
===Peer nodes===
  
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{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction @ InterSciWiki]
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{{col-break}}
 
 
* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz]
 
* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz]
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* [http://mathweb.org/wiki/Exclusive_disjunction Exclusive Disjunction @ MathWeb Wiki]
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* [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis]
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* [http://netknowledge.org/wiki/Exclusive_disjunction Exclusive Disjunction @ NetKnowledge]
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* [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity]
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{{col-break}}
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* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta]
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* [http://wiki.oercommons.org/mediawiki/index.php/Exclusive_disjunction Exclusive Disjunction @ OER Commons]
 
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* [http://p2pfoundation.net/Exclusive_Disjunction Exclusive Disjunction @ P2P Foundation]
 
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* [http://semanticweb.org/wiki/Exclusive_disjunction Exclusive Disjunction @ SemanticWeb]
 
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===Logical operators===
 
===Logical operators===
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{{col-break}}
 
{{col-break}}
 
* [[Inquiry]]
 
* [[Inquiry]]
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* [[Dynamics of inquiry]]
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* [[Semeiotic]]
 
* [[Logic of information]]
 
* [[Logic of information]]
 
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* [[Pragmatic maxim]]
 
* [[Pragmatic maxim]]
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* [[Pragmatic theory of truth]]
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* [[Truth theory]]
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* [[Semeiotic]]
 
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* [[Semiotic information]]
 
 
{{col-end}}
 
{{col-end}}
  
 
===Related articles===
 
===Related articles===
  
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
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{{col-begin}}
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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{{col-break}}
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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{{col-break}}
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* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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{{col-end}}
  
 
==Document history==
 
==Document history==
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Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
 
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
  
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{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction], [http://intersci.ss.uci.edu/ InterSciWiki]
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{{col-break}}
 
 
* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz]
 
* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz]
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* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Beta Wikiversity]
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* [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://ref.subwiki.org/ Subject Wikis]
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* [http://www.getwiki.net/-Exclusive_Disjunction Exclusive Disjunction], [http://www.getwiki.net/ GetWiki]
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* [http://wikinfo.org/w/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/w/ Wikinfo]
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{{col-break}}
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* [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://en.wikiversity.org/ Wikiversity]
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* [http://www.wikinfo.org/index.php/Exclusive_disjunction Exclusive Disjunction], [http://www.wikinfo.org/ Wikinfo]
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* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [http://www.textop.org/wiki/index.php?title=Exclusive_disjunction Exclusive Disjunction], [http://www.textop.org/wiki/ Textop Wiki]
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* [http://en.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia]
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* [http://en.wikipedia.org/w/index.php?title=Exclusive_or&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia]
 
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[[Category:Inquiry]]
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[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
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[[Category:Charles Sanders Peirce]]
 
[[Category:Computer Science]]
 
[[Category:Computer Science]]
 
[[Category:Formal Languages]]
 
[[Category:Formal Languages]]

Latest revision as of 01:45, 31 October 2015

☞ This page belongs to resource collections on Logic and Inquiry.

Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.

The truth table of \(p ~\operatorname{XOR}~ q,\) also written \(p + q~\!\) or \(p \ne q,\!\) appears below:


\(\text{Exclusive Disjunction}\!\)
\(p\!\) \(q\!\) \(p ~\operatorname{XOR}~ q\)
\(\operatorname{F}\) \(\operatorname{F}\) \(\operatorname{F}\)
\(\operatorname{F}\) \(\operatorname{T}\) \(\operatorname{T}\)
\(\operatorname{T}\) \(\operatorname{F}\) \(\operatorname{T}\)
\(\operatorname{T}\) \(\operatorname{T}\) \(\operatorname{F}\)


The following equivalents may then be deduced:

\(\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\[6pt] & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\[6pt] & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\)

Syllabus

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Logical operators

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Relational concepts

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Information, Inquiry

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Related articles

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Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

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