Exclusive Or is Negation of Biconditional

Theorem

Exclusive or is equivalent to the negation of the biconditional:

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


Proof

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

$\blacksquare$


Sources

  • 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.1$
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