Definition:Exclusive Or

Definition
Exclusive or is a binary connective which can be written symbolically as $p \oplus q$ whose behaviour is as follows:


 * $p \oplus q$

means:
 * Either $p$ is true or $q$ is true but not both.

or symbolically:
 * $p \oplus q := \paren {p \lor q} \land \neg \paren {p \land q}$

where $\land$ denotes the and operator and $\lor$ denotes the or operator.

There is no standard symbol for this, but the one shown above is seen commonly enough to be adopted as standard for this site.

Also known as
This usage of or, that disallows the case where both disjuncts are true, is also called:
 * exclusive disjunction
 * logical inequality
 * non-equivalence
 * symmetric difference
 * the alternative function
 * aut (from the Latin), pronounced out.

Some sources refer to this as the strong or, where the weak or is used in the sense of the inclusive or.

In natural language, when it is necessary to be precise about the nature of the term being used, the phrase but not both is often employed.

Some sources give the symbol as $\underline \lor$ or $\not \equiv$

Also see

 * Exclusive Or is Negation of Biconditional
 * Non-Equivalence