Difference between revisions of "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.
 
'''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 <math>p ~\operatorname{XOR}~ q</math> (also written as <math>p + q\!</math> or <math>p \ne q\!</math>) is as follows:
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The [[truth table]] of <math>p ~\operatorname{XOR}~ q,</math> also written as <math>p + q\!</math> or <math>p \ne q,\!</math> is shown below:
  
 
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Revision as of 11:12, 15 May 2012

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 as \(p + q\!\) or \(p \ne q,\!\) is shown 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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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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