# 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

- 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 $1.1.1$