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.

There is no standard symbol for this, but the one shown above is commonly seen.

The operation $\oplus$ is called (from the Latin) aut (prounounced out).

This usage of or, that disallows the case where both disjuncts are true, is also called:

• exclusive disjunction;
• logical inequality;
• non-equivalence;
• symmetric difference.

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 material equivalence operator.

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

Truth Table

The truth table of $p \oplus q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \oplus q & p \iff q \\ \hline F&F&F&T\\ F&T&T&F\\ T&F&T&F\\ T&T&F&T\\ \hline \end{array}$

Notational Variants

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

$p \oplus q$	sometimes called o-plus
$p\ \mathsf{XOR} \ q$
$p + q$
$p \not \Leftrightarrow q$
$p \not \equiv q$
$p \ne q$
$p \ \dot \lor \ q$

Sources

• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$: Exercise $1.1.1$