Exclusive Or as Conjunction of Disjunctions/Proof 2
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Theorem
- $p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$
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.
$\begin{array}{|ccc||ccccccccc|} \hline
p & \oplus & q & (p & \lor & q) & \land & (\neg & p & \lor & \neg & q) \\
\hline
F & F & F & F & F & F & F & T & F & T & T & F \\
F & T & T & F & T & T & T & T & F & T & F & T \\
T & T & F & T & T & F & T & F & T & T & T & F \\
T & F & T & T & T & T & F & F & T & F & F & T \\
\hline
\end{array}$
$\blacksquare$