I'm trying to learn some alghoritms of boolean logic and I encountered a problem wich i don't understand.
There is a expression and I don't understand how to simplify it.
$$(A \wedge \neg B) \vee(\neg A \wedge B) \implies(A \vee B) \wedge (\neg A \vee \neg B)$$
How am I supposed to simplify left part of expression to the right one?
Thanks for your help!
Best Answer
\begin{align*} (A \wedge \neg B) \vee(\neg A \wedge B) &\implies [(A \wedge \neg B)\vee\neg A]\wedge[(A \wedge \neg B)\vee B]\\ &\implies [(A\vee\neg A)\wedge(\neg B\vee \neg A)]\wedge[(A\vee B)\wedge(\neg B\vee B)]\\ &\implies [T\wedge(\neg B\vee \neg A)]\wedge[(A\vee B)\wedge T]\\ &\implies (\neg B\vee \neg A)\wedge(A\vee B)\\ &\implies(A \vee B) \wedge (\neg A \vee \neg B) \end{align*}