Exclusive Or is Associative

Theorem
Exclusive or is associative:
 * $p \oplus \paren {q \oplus r} \dashv \vdash \paren {p \oplus q} \oplus r$

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

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