# Exclusive disjunction

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 $p+q\!$ or $p\neq q,\!$ appears below:

 $p\!$ $q\!$ $p~\operatorname {XOR} ~q$ $\operatorname {F}$ $\operatorname {F}$ $\operatorname {F}$ $\operatorname {F}$ $\operatorname {T}$ $\operatorname {T}$ $\operatorname {T}$ $\operatorname {F}$ $\operatorname {T}$ $\operatorname {T}$ $\operatorname {T}$ $\operatorname {F}$ The following equivalents may then be deduced:

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

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