Exclusive Or is Commutative

Theorem

Exclusive or is commutative:

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

Proof 1

By the tableau method of natural deduction:

$p \oplus q \vdash q \oplus p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \oplus q$	Premise	(None)
2		1	$\left({p \lor q} \right) \land \neg \left({p \land q}\right)$	Sequent Introduction	1	Definition of Exclusive Or
3		1	$\left({q \lor p} \right) \land \neg \left({p \land q}\right)$	Sequent Introduction	2	Disjunction is Commutative
4		1	$\left({q \lor p} \right) \land \neg \left({q \land p}\right)$	Sequent Introduction	3	Conjunction is Commutative
5		1	$q \oplus p$	Sequent Introduction	4	Definition of Exclusive Or

By the tableau method of natural deduction:

$q \oplus p \vdash p \oplus q$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$q \oplus p$	Premise	(None)
2		1	$\left({q \lor p} \right) \land \neg \left({q \land p}\right)$	Sequent Introduction	1	Definition of Exclusive Or
3		1	$\left({q \lor p} \right) \land \neg \left({p \land q}\right)$	Sequent Introduction	2	Conjunction is Commutative
4		1	$\left({p \lor q} \right) \land \neg \left({p \land q}\right)$	Sequent Introduction	3	Disjunction is Commutative
5		1	$p \oplus q$	Sequent Introduction	4	Definition of Exclusive Or

$\blacksquare$

Proof 2

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.

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

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$