Definition:Exclusive Or

Definition
Exclusive Or is a binary connective which can be written symbolically as $$p \not \Leftrightarrow q$$ whose behaviour is as follows:


 * $$p \not \Leftrightarrow q$$ means "either $$p$$ is true or $$q$$ is true but not both."

There is no standard symbol for this, but this one is common and intuitively obvious.

The operation $$\not \Leftrightarrow$$ 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.

Complement
The complement of $$\not \Leftrightarrow$$ is the material equivalence operator.

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

Truth Table
The truth table of $$p\not \Leftrightarrow q$$ and its complement is as follows:

Boolean Interpretation
From the above, we see that the boolean interpretations for $$p \not \Leftrightarrow q$$ are:

Notational Variants
Alternative symbols that mean the same thing as $$p \not \Leftrightarrow q$$ are also encountered:

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