Exclusive Or as Conjunction of Disjunctions/Proof 2

Theorem

$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$


Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.


$\begin{array}{|ccc||ccccccccc|} \hline p & \oplus & q & (p & \lor & q) & \land & (\neg & p & \lor & \neg & q) \\ \hline \F & \F & \F & \F & \F & \F & \F & \T & \F & \T & \T & \F \\ \F & \T & \T & \F & \T & \T & \T & \T & \F & \T & \F & \T \\ \T & \T & \F & \T & \T & \F & \T & \F & \T & \T & \T & \F \\ \T & \F & \T & \T & \T & \T & \F & \F & \T & \F & \F & \T \\ \hline \end{array}$

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.