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: