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 this one 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
Alternative symbols that mean the same thing as $$p \oplus q$$ are also encountered:

Other symbols that are commonly seen are:
 * $$p\ \texttt{XOR}\ q$$;
 * $$p + q$$;
 * $$p \not \Leftrightarrow q$$;
 * $$p \not \equiv q$$;
 * $$p \ne q$$;
 * $$p \dot \lor q$$.