# Thread: how are these two statments logically equivalent?

1. ## how are these two statments logically equivalent?

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

2. ## Re: how are these two statments logically equivalent?

the second statement should be written as

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

this follows from the first statement using distributive law

Boolean algebra (logic) - Wikipedia, the free encyclopedia

3. ## Re: how are these two statments logically equivalent?

But wouldn't that distribute as $[(p \vee q ) \wedge ( \neg q \vee q)] \wedge [(p \vee \neg p) \wedge ( \neg q \vee \neg p )]$ which isn't the same as the prior step

4. ## Re: how are these two statments logically equivalent?

yes you are right. you have further used the distributive law.
further you can see that
$q\vee \neg q \;\;\mbox{and}\;\; p\vee \neg p$

are tautologies. so we can further simplify as

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

5. ## Re: 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)

6. ## Re: how are these two statments logically equivalent?

do you understand now ?