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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