# Exclusive Or is Self-Inverse

## Theorem

$\paren {p \oplus q} \oplus q \dashv \vdash p$

where $\oplus$ denotes the exclusive or operator.

## Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective on the left hand side match those for $p$ on the right hand side for all boolean interpretations:

$\begin{array}{|ccccc||c|} \hline (p & \oplus & q) & \oplus & q & p \\ \hline \F & \F & \F & \F & \F & \F \\ \F & \T & \T & \F & \T & \F \\ \T & \T & \F & \T & \F & \T \\ \T & \F & \T & \T & \T & \T \\ \hline \end{array}$

$\blacksquare$