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''', '''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>
 
|-
 
|-
−
| 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===
 +
 +
* [[Inquiry Live]]
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* [[Logic Live]]
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 +
===Peer nodes===
 +
 +
* [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]
 +
* [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}}
 +
{{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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{|
+
 
−
| valign=top |
+
{{col-begin}}
 +
{{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]]
−
* [[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 |
+
* [[Minimal negation operator]]
 +
* [[Multigrade operator]]
 +
* [[Parametric operator]]
 
* [[Peirce's law]]
 
* [[Peirce's law]]
 +
{{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===
 +
 
 +
{{col-begin}}
 +
{{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]]
 +
{{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}}
 +
 
 +
===Information, Inquiry===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
 +
{{col-break}}
 +
* [[Semeiotic]]
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* [[Logic of information]]
 +
{{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===
 +
 
 +
{{col-begin}}
 +
{{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]
 +
{{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]
 +
{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 +
{{col-end}}
 +
 
 +
==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.
 +
 
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Exclusive_disjunction Exclusive Disjunction], [http://intersci.ss.uci.edu/ InterSciWiki]
 +
* [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction], [http://mywikibiz.com/ MyWikiBiz]
 +
* [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://ref.subwiki.org/ Subject Wikis]
 +
* [http://wikinfo.org/w/index.php/Exclusive_disjunction Exclusive Disjunction], [http://wikinfo.org/w/ Wikinfo]
 +
* [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://en.wikiversity.org/ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction], [http://beta.wikiversity.org/ Wikiversity Beta]
 +
* [http://en.wikipedia.org/w/index.php?title=Exclusive_disjunction&oldid=75153068 Exclusive Disjunction], [http://en.wikipedia.org/ Wikipedia]
 +
 
 +
[[Category:Inquiry]]
 +
[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
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[[Category:Charles Sanders Peirce]]
 +
[[Category:Computer Science]]
 +
[[Category:Formal Languages]]
 +
[[Category:Formal Sciences]]
 +
[[Category:Formal Systems]]
 +
[[Category:Linguistics]]
 +
[[Category:Logic]]
 +
[[Category:Mathematics]]
 +
[[Category:Philosophy]]
 +
[[Category:Semiotics]]

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

Focal nodes

Peer nodes

Logical operators

Template:Col-breakTemplate:Col-breakTemplate:Col-end

Related topics

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Relational concepts

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Information, Inquiry

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Related articles

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

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.

Retrieved from "https://mywikibiz.com/index.php?title=Exclusive_disjunction&oldid=466806"