Hi,
I have been trying to work around the following question, but can't seem to get it right.
(P ^ ¬Q) V (¬P ^ Q) <=> ¬(P ^ Q) ^ (P VQ)
Any help is highly appreciated.
$\displaystyle \begin{gathered} (P \wedge \neg Q) \vee (\neg P \wedge Q) \Leftrightarrow \hfill \\ \left[ {(P \wedge \neg Q) \vee \neg P} \right] \wedge \left[ {(P \wedge \neg Q) \vee Q} \right] \Leftrightarrow \hfill \\ \left[ {\left( {P \vee \neg P} \right) \wedge \left( {\neg Q \vee \neg P} \right)} \right] \wedge \left[ {\left( {P \vee Q} \right) \wedge \left( {\neg Q \vee Q} \right)} \right] \Leftrightarrow \hfill \\ \neg \left( {P \wedge Q} \right) \wedge (P \vee Q) \hfill \\\end{gathered}$
I have tried doing this:
(P ^ ¬Q) V (¬P ^ Q) <=> ¬(P ^ Q) ^ (P VQ)
Swap sides
¬(P ^ Q) ^ (P VQ) <=> (P ^ ¬Q) V (¬P ^ Q)
Hence, since ¬(P ^Q) <=> ¬P V ¬Q (using De Morgan's Laws)
This gives us:(¬P V ¬Q) ^ (P V Q)
And in turn, (¬P V ¬Q) ^ (P V Q)<=> (P ^ ¬Q) V (¬P ^ Q)
However, I can't seem to figure out where to go from here.