# 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 $$p ~\operatorname{XOR}~ q,$$ also written $$p + q~\!$$ or $$p \ne q,\!$$ 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:

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

## Syllabus

### Logical operators

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## 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.