Difference between revisions of "Exclusive disjunction"
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<font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. | <font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. | ||
− | '''Exclusive disjunction''', also known as '''logical inequality''' or '''symmetric difference''', is an | + | '''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 | + | 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> | ||
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style=" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%" |
− | |+ | + | |+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math> |
− | |- style="background:# | + | |- style="height:40px; background:#f0f0ff" |
− | + | | style="width:33%" | <math>p\!</math> | |
− | + | | style="width:33%" | <math>q\!</math> | |
− | + | | style="width:33%" | <math>p ~\operatorname{XOR}~ q</math> | |
|- | |- | ||
− | | F || F || F | + | | <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math> |
|- | |- | ||
− | | F || T || T | + | | <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math> |
|- | |- | ||
− | | T || F || T | + | | <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math> |
|- | |- | ||
− | | T || T || F | + | | <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math> |
|} | |} | ||
<br> | <br> | ||
− | The following equivalents | + | The following equivalents may then be deduced: |
− | + | {| align="center" cellspacing="10" width="90%" | |
− | p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) | + | | |
− | \\ | + | <math>\begin{matrix} |
− | & = & (p \lor q) & \land & (\lnot p \lor \lnot q) | + | 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) | & = & (p \lor q) & \land & \lnot (p \land q) | ||
\end{matrix}</math> | \end{matrix}</math> | ||
+ | |} | ||
==Syllabus== | ==Syllabus== | ||
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===Focal nodes=== | ===Focal nodes=== | ||
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* [[Inquiry Live]] | * [[Inquiry Live]] | ||
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* [[Logic Live]] | * [[Logic Live]] | ||
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===Peer nodes=== | ===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] | * [http://mywikibiz.com/Exclusive_disjunction Exclusive Disjunction @ MyWikiBiz] | ||
− | * [http:// | + | * [http://ref.subwiki.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Subject Wikis] |
− | + | * [http://en.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity] | |
− | + | * [http://beta.wikiversity.org/wiki/Exclusive_disjunction Exclusive Disjunction @ Wikiversity Beta] | |
− | * [http:// | ||
− | * [http:// | ||
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===Logical operators=== | ===Logical operators=== | ||
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===Related articles=== | ===Related articles=== | ||
− | * [http:// | + | {{col-begin}} |
− | + | {{col-break}} | |
− | * [http:// | + | * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] |
− | + | * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] | |
− | * [http:// | + | * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] |
− | + | {{col-break}} | |
− | * [http:// | + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] |
− | + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] | |
− | * [http:// | + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] |
− | + | {{col-break}} | |
− | * [http:// | + | * [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:// | + | * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] |
− | + | {{col-end}} | |
− | * [http:// | ||
==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. | ||
− | + | * [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] | * [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://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] | * [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]] | [[Category:Inquiry]] |
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:
\(p\!\) | \(q\!\) | \(p ~\operatorname{XOR}~ q\) |
\(\operatorname{F}\) | \(\operatorname{F}\) | \(\operatorname{F}\) |
\(\operatorname{F}\) | \(\operatorname{T}\) | \(\operatorname{T}\) |
\(\operatorname{T}\) | \(\operatorname{F}\) | \(\operatorname{T}\) |
\(\operatorname{T}\) | \(\operatorname{T}\) | \(\operatorname{F}\) |
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
- Exclusive Disjunction @ InterSciWiki
- Exclusive Disjunction @ MyWikiBiz
- Exclusive Disjunction @ Subject Wikis
- Exclusive Disjunction @ Wikiversity
- Exclusive Disjunction @ Wikiversity Beta
Logical operators
Template:Col-breakTemplate:Col-breakTemplate:Col-endRelated topics
- Propositional calculus
- Sole sufficient operator
- Truth table
- Universe of discourse
- Zeroth order logic
Relational concepts
Information, Inquiry
Related articles
- Differential Logic : Introduction
- Differential Propositional Calculus
- Differential Logic and Dynamic Systems
- Prospects for Inquiry Driven Systems
- Introduction to Inquiry Driven Systems
- Inquiry Driven Systems : Inquiry Into Inquiry
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.
- Exclusive Disjunction, InterSciWiki
- Exclusive Disjunction, MyWikiBiz
- Exclusive Disjunction, Subject Wikis
- Exclusive Disjunction, Wikinfo
- Exclusive Disjunction, Wikiversity
- Exclusive Disjunction, Wikiversity Beta
- Exclusive Disjunction, Wikipedia