Exclusive disjunction

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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.

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:

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

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Logical operators

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• Exclusive disjunction
• Logical conjunction
• Logical disjunction
• Logical equality

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• Logical implication
• Logical NAND
• Logical NNOR
• Negation

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Related topics

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• Ampheck
• Boolean domain
• Boolean function
• Boolean-valued function
• Differential logic

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• Logical graph
• Minimal negation operator
• Multigrade operator
• Parametric operator
• Peirce's law

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• Propositional calculus
• Sole sufficient operator
• Truth table
• Universe of discourse
• Zeroth order logic

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Relational concepts

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• Continuous predicate
• Hypostatic abstraction
• Logic of relatives
• Logical matrix

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• Relation
• Relation composition
• Relation construction
• Relation reduction

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• Relation theory
• Relative term
• Sign relation
• Triadic relation

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Information, Inquiry

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• Inquiry
• Dynamics of inquiry

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• Semeiotic
• Logic of information

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• Descriptive science
• Normative science

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• Pragmatic maxim
• Truth theory

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Related articles

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• Cactus Language
• Futures Of Logical Graphs
• Propositional Equation Reasoning Systems

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• Differential Logic : Introduction
• Differential Propositional Calculus
• Differential Logic and Dynamic Systems

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• Prospects for Inquiry Driven Systems
• Introduction to Inquiry Driven Systems
• Inquiry Driven Systems : Inquiry Into Inquiry

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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.

• Exclusive Disjunction, InterSciWiki
• Exclusive Disjunction, MyWikiBiz
• Exclusive Disjunction, Subject Wikis
• Exclusive Disjunction, Wikinfo
• Exclusive Disjunction, Wikiversity
• Exclusive Disjunction, Wikiversity Beta
• Exclusive Disjunction, Wikipedia