Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd.

It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR excludes that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".

It is symbolized by the prefix operator ${\displaystyle J}$ and by the infix operators XOR (/ˌɛks ˈɔːr/, /ˌɛks ˈɔː/, /ˈksɔːr/ or /ˈksɔː/), EOR, EXOR, ${\displaystyle {\dot {\vee }}}$, ${\displaystyle {\overline {\vee }}}$, ${\displaystyle {\underline {\vee }}}$, ⩛, ${\displaystyle \oplus }$, ${\displaystyle \nleftrightarrow }$, and ${\displaystyle \not \equiv }$.

The truth table of ${\displaystyle A\nleftrightarrow B}$ shows that it outputs true whenever the inputs differ:

Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction ${\displaystyle p\nleftrightarrow q}$, also denoted by ${\displaystyle p\operatorname {?} q}$ or ${\displaystyle Jpq}$, can be expressed in terms of the logical conjunction ("logical and", ${\displaystyle \land }$), the disjunction ("logical or", ${\displaystyle \vee }$), and the negation (${\displaystyle \neg }$) as follows:

The exclusive disjunction ${\displaystyle p\nleftrightarrow q}$ can also be expressed in the following way:

This representation of XOR may be found useful when constructing a circuit or network, because it has only one ${\displaystyle \lnot }$ operation and small number of ${\displaystyle \land }$ and ${\displaystyle \lor }$ operations. A proof of this identity is given below:

It is sometimes useful to write ${\displaystyle p\nleftrightarrow q}$ in the following way: