What is "exclusive or"

n. 1 (context logic computing English) exclusive disjunction: the use of (term: or) to indicate that of two predicates, one is true and one is false (without specifying which is which); (non-gloss definition: contrasted with inclusive or, which does not imply that one must be false.) 2 (context logic computing more generally English) exclusive disjunction: the use of (term: or) to indicate that of two or more predicates, an odd number are true (without specifying which or how many); (non-gloss definition: contrasted with inclusive or, which indicates only that one or more is true.) 3 (context logic computing English) An exclusive disjunction; the result of applying the above-described exclusive or to two or more predicates; (non-gloss definition: contrasted with an inclusive or, which is the result of applying an inclusive or.)

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Venn diagram of $\scriptstyle A \oplus B$
but not is

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Venn diagram of $\scriptstyle A \oplus B \oplus C$
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Exclusive disjunction or exclusive or is a logical operation that outputs true only when inputs differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR , EOR, EXOR, ⊻, ⊕, ↮, and ≢. The negation of XOR is logical biconditional, which outputs true only when both inputs are the same.

It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B".

More generally, XOR is true only when an odd number of inputs are true. A chain of XORs—a XOR b XOR c XOR d (and so on)—is true whenever an odd number of the inputs are true and is false whenever an even number of inputs are true.