Exclusive Or as Conjunction of Disjunctions

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Theorem

$p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$


Proof 1

\(\displaystyle p \oplus q\) \(\dashv \vdash\) \(\displaystyle \left({p \lor q}\right) \land \neg \left({p \land q}\right)\) Definition of Exclusive Or
\(\displaystyle \) \(\dashv \vdash\) \(\displaystyle \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)\) De Morgan's Laws: Disjunction of Negations

$\blacksquare$


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.


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

$\blacksquare$


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