Exclusive Or as Conjunction of Disjunctions/Proof 1

Theorem

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


Proof

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

$\blacksquare$

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