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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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.