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 $p\neq q,\!$ 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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