Exclusive Or as Disjunction of Conjunctions/Proof 1

From ProofWiki
Jump to: navigation, search

Theorem

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


Proof

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

$\blacksquare$