The exclusive disjunction is a truth-functional operator in logic which is equivalent to the English wording “either“ or”. It is used to express a disjunction in which the connected propositions or formulae cannot both be true.
For example, if P or Q are true, but not both, we may say formally:
P ⊕ Q
This may be read as “Either P or Q” or “it is the case that P or Q, but not both”.
In symbolic logic, the exclusive disjunction symbol ( ⊕ ) is used to symbolize this type of disjunction. Sometimes, the capitalized term XOR (meaning “exclusive or”) is used.
Some logical systems do not include the ⊕ symbol in their formal language. In these systems, exclusive disjunctions must be formulated using other operators. The following is an alternative expression of an exclusive disjunction:
(P ∨ Q) ∧ ¬(P & Q)
The following table illustrates the possible truth values of P ⊕ Q, given each possible valuation of its terms, P and Q. Note that unlike an inclusive disjunction, P ⊕ Q is true if, and only if, exactly one of its connected terms is true.
|P||Q||P ⊕ Q|