Exclusive Or as Disjunction of Conjunctions/Proof 1

Theorem

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

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle p \oplus q$	$\dashv \vdash$	$\displaystyle $	$\displaystyle \neg \left ({p \iff q}\right)$	$\displaystyle $	$\displaystyle $	Exclusive Or is Negation of Biconditional
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\dashv \vdash$	$\displaystyle $	$\displaystyle \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)$	$\displaystyle $	$\displaystyle $	Non-Equivalence as Disjunction of Conjunctions

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p \oplus q\)	\(\dashv \vdash\)	\(\displaystyle \)	\(\displaystyle \neg \left ({p \iff q}\right)\)	\(\displaystyle \)	\(\displaystyle \)	Exclusive Or is Negation of Biconditional
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\dashv \vdash\)	\(\displaystyle \)	\(\displaystyle \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)\)	\(\displaystyle \)	\(\displaystyle \)	Non-Equivalence as Disjunction of Conjunctions