Exclusive Or as Disjunction of Conjunctions/Proof 1

Theorem

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

Proof

\(\ds p \oplus q\)

\(\dashv \vdash\)

\(\ds \neg \left ({p \iff q}\right)\)

Exclusive Or is Negation of Biconditional

\(\ds \)

\(\dashv \vdash\)

\(\ds \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)\)

Non-Equivalence as Disjunction of Conjunctions

$\blacksquare$