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
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.