# Exclusive Or as Disjunction of Conjunctions/Proof 1

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## Theorem

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

## Proof

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

$\blacksquare$