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It's not apparent how these are logically equivalent, could someone please elaborate?

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- Oct 3rd 2011, 07:46 PMJskidhow are these two statments logically equivalent?
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It's not apparent how these are logically equivalent, could someone please elaborate? - Oct 3rd 2011, 08:00 PMissacnewtonRe: how are these two statments logically equivalent?
the second statement should be written as

$\displaystyle ((p\wedge \neg q)\vee q)\wedge((p\wedge \neg q)\vee \neg p)$

this follows from the first statement using distributive law

Boolean algebra (logic) - Wikipedia, the free encyclopedia - Oct 3rd 2011, 08:23 PMJskidRe: how are these two statments logically equivalent?
But wouldn't that distribute as $\displaystyle [(p \vee q ) \wedge ( \neg q \vee q)] \wedge [(p \vee \neg p) \wedge ( \neg q \vee \neg p )]$ which isn't the same as the prior step

- Oct 3rd 2011, 09:22 PMissacnewtonRe: how are these two statments logically equivalent?
yes you are right. you have further used the distributive law.

further you can see that

$\displaystyle q\vee \neg q \;\;\mbox{and}\;\; p\vee \neg p $

are tautologies. so we can further simplify as

$\displaystyle (p\vee q)\wedge (\neg p \vee \neg q)$ - Oct 3rd 2011, 09:44 PMJskidRe: how are these two statments logically equivalent?
I couldn't see that b/c I was trying to think of how to apply the distributive law differently. I never occured to me that I needed to do another step (i.e. remove tautologies)

- Oct 3rd 2011, 09:49 PMissacnewtonRe: how are these two statments logically equivalent?
do you understand now ?