Exclusive Or is Negation of Biconditional

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Theorem

Exclusive or is equivalent to the negation of the biconditional:

$p \oplus q \dashv \vdash \neg \paren {p \iff q}$


Proof

\(\displaystyle p \oplus q\) \(\dashv \vdash\) \(\displaystyle \paren {p \lor q} \land \neg \paren {p \land q}\) Definition of Exclusive Or
\(\displaystyle \) \(\dashv \vdash\) \(\displaystyle \neg \paren {p \iff q}\) Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction

$\blacksquare$


Sources