Exclusive Or Properties

Theorems
Exclusive or is commutative:


 * $$p \oplus q \dashv \vdash q \oplus p$$

Exclusive or is associative:


 * $$p \oplus \left({q \oplus r}\right) \dashv \vdash \left({p \oplus q}\right) \oplus r$$

Exclusive or destroys copies of itself:


 * $$p \oplus p \dashv \vdash \bot$$

Proof by Natural deduction
Commutativity is proved by the Tableau method:

$$q \oplus p \vdash p \oplus q$$ is proved similarly.

Proof of associativity by natural deduction is just too tedious to be considered.

Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.

As can be seen by inspection, in all cases the truth values under the main connectives match for all models.

$$\begin{array}{|ccc||ccc|} \hline p & \oplus & q & q & \oplus & p \\ \hline F & F & F & F & F & F \\ F & T & T & T & T & F \\ T & T & F & F & T & T \\ T & F & T & T & F & T \\ \hline \end{array}$$

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

$$\begin{array}{|ccc|} \hline p & \oplus & p \\ \hline F & F & F \\ T & F & T \\ \hline \end{array}$$