Exclusive Or as Disjunction of Conjunctions/Proof 2

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Theorem

$p \oplus q \dashv \vdash \paren {\neg p \land q} \lor \paren {p \land \neg q}$


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.


$\begin{array}{|ccc||ccccccccc|} \hline p & \oplus & q & (\neg & p & \land & q) & \lor & (p & \land & \neg & q) \\ \hline F & F & F & T & F & F & F & F & F & F & T & F \\ F & T & T & T & F & T & T & T & F & F & F & T \\ T & T & F & F & T & F & F & T & T & T & T & F \\ T & F & T & F & T & F & T & F & T & F & F & T \\ \hline \end{array}$

$\blacksquare$