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Exclusive disjunction (view source)

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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''', '''p &oplus; 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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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
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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:
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|+ '''Exclusive Disjunction'''
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|- style="background:paleturquoise"
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<br>
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! style="width:15%" | p
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! style="width:15%" | q
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
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! style="width:15%" | p XOR q
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|+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math>
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|- style="height:40px; background:#f0f0ff"
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| style="width:33%" | <math>p\!</math>
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| style="width:33%" | <math>q\!</math>
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| style="width:33%" | <math>p ~\operatorname{XOR}~ q</math>
 
|-
 
|-
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| 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>
 
|}
 
|}
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<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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|}
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==Syllabus==
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===Focal nodes===
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* [[Inquiry Live]]
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* [[Logic Live]]
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===Peer nodes===
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* [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction @ InterSciWiki]
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* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis]
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* [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta]
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==See also==
   
===Logical operators===
 
===Logical operators===
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{|
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| valign=top |
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{{col-begin}}
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{{col-break}}
 
* [[Exclusive disjunction]]
 
* [[Exclusive disjunction]]
 
* [[Logical conjunction]]
 
* [[Logical conjunction]]
 
* [[Logical disjunction]]
 
* [[Logical disjunction]]
 
* [[Logical equality]]
 
* [[Logical equality]]
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| valign=top |
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{{col-break}}
 
* [[Logical implication]]
 
* [[Logical implication]]
 
* [[Logical NAND]]
 
* [[Logical NAND]]
 
* [[Logical NNOR]]
 
* [[Logical NNOR]]
 
* [[Logical negation|Negation]]
 
* [[Logical negation|Negation]]
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|}
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{{col-end}}
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===Related topics===
 
===Related topics===
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{|
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| valign=top |
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{{col-begin}}
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{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
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* [[Boolean algebra]]
   
* [[Boolean domain]]
 
* [[Boolean domain]]
 
* [[Boolean function]]
 
* [[Boolean function]]
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| valign=top |
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* [[Boolean-valued function]]
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* [[Boolean logic]]
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* [[Differential logic]]
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* [[Laws of Form]]
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{{col-break}}
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* [[Logic gate]]
   
* [[Logical graph]]
 
* [[Logical graph]]
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| valign=top |
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* [[Minimal negation operator]]
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* [[Multigrade operator]]
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* [[Parametric operator]]
 
* [[Peirce's law]]
 
* [[Peirce's law]]
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{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
 
* [[Sole sufficient operator]]
 
* [[Sole sufficient operator]]
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* [[Truth table]]
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* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
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|}
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{{col-end}}
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===Relational concepts===
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{{col-begin}}
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{{col-break}}
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* [[Continuous predicate]]
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* [[Hypostatic abstraction]]
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* [[Logic of relatives]]
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* [[Logical matrix]]
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{{col-break}}
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* [[Relation (mathematics)|Relation]]
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* [[Relation composition]]
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* [[Relation construction]]
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* [[Relation reduction]]
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{{col-break}}
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* [[Relation theory]]
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* [[Relative term]]
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* [[Sign relation]]
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* [[Triadic relation]]
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{{col-end}}
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===Information, Inquiry===
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{{col-begin}}
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{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
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{{col-break}}
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* [[Semeiotic]]
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* [[Logic of information]]
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{{col-break}}
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* [[Descriptive science]]
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* [[Normative science]]
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{{col-break}}
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* [[Pragmatic maxim]]
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* [[Truth theory]]
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{{col-end}}
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===Related articles===
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{{col-begin}}
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{{col-break}}
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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://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://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://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://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://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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{{col-end}}
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==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.
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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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* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://ref.subwiki.org/ Subject Wikis]
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* [http://wikinfo.org/w/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/w/ Wikinfo]
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* [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://en.wikiversity.org/ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta]
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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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[[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]]
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[[Category:Computer Science]]
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[[Category:Formal Languages]]
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[[Category:Formal Sciences]]
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[[Category:Formal Systems]]
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[[Category:Linguistics]]
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[[Category:Logic]]
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[[Category:Mathematics]]
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[[Category:Philosophy]]
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[[Category:Semiotics]]
Jon Awbrey
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