Quickly add a free MyWikiBiz directory listing!

# 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 $$p \ne q,\!$$ 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

 Exclusive disjunction Logical conjunction Logical disjunction Logical equality

## 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.