# Exclusive Or is Commutative

## Theorem

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

## 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.

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

$\blacksquare$