# Exclusive disjunction

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 ${\displaystyle p~{\text{XOR}}~q}$ , also written ${\displaystyle p+q}$ or ${\displaystyle p\neq q}$ appears below:

 ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p~{\text{XOR}}~q}$ ${\displaystyle {\text{F}}}$ ${\displaystyle {\text{F}}}$ ${\displaystyle {\text{F}}}$ ${\displaystyle {\text{F}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{F}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{T}}}$ ${\displaystyle {\text{F}}}$

The following equivalents may then be deduced:

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

## Document history

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.