Exclusive Or as Conjunction of Disjunctions/Proof 1
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Theorem
- $p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$
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 |
$\blacksquare$