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:
 * $p \oplus q := \left({p \lor q} \right) \land \neg \left({p \land q}\right)$

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 commonly enough seen to be adopted as standard for this site.

Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \oplus \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \oplus \mathbf B}\right)_\mathcal M = \begin{cases}

F & : \mathbf A_\mathcal M = \mathbf B_\mathcal M \\ T & : \text {otherwise} \end{cases}$

Complement
The complement of $\oplus$ is the biconditional operator.

See Non-Equivalence for the proofs of some results relating these operators.

Notational Variants
Various symbols are encountered that denote the concept of exclusive or:

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
 * aut (from the Latin), prounounced out.

Also see

 * Exclusive Or is Negation of Biconditional
 * Non-Equivalence