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
\(\ds p \oplus q\) | \(\dashv \vdash\) | \(\ds \paren {p \lor q} \land \neg \paren {p \land q}\) | 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 $1.1.1$