# Help with logical equivalences

• August 28th 2010, 12:36 PM
learningguy
Help with logical equivalences
This problem asks me to use the p→q ≡~pvq and p↔q≡(~pvq) ^ (~qvp) to rewrite the given statement forms without using the symbol → or ↔.

The problem is (p→(q→r)) ↔ ((p^q)→r)

Using a bunch of truth tables and kinda just guessing I was able to come up with ~(~pv(~qvr)) v (~(p^q)vr) or (~pv(~qvr)) v ~(~(p^q)vr), both of which would work I think but I am not 100% sure they are correct so if someone wants to double check them that would be good. I think there is a quicker way to do this but the logical equivalence that I should use isn't really obvious to me just looking at a problem that complex. I guess I'm just asking for some tips on how to do this so any help would be appreciated.
• August 28th 2010, 12:40 PM
Ackbeet

$(p\to(q\to r))\iff((p\land q)\to r),$ and convert one step at a time. The first step would be

$(p\to(\neg q\vee r))\iff(\neg(p\land q)\vee r).$

Can you continue?
• August 28th 2010, 12:48 PM
Isomorphism
Quote:

Originally Posted by learningguy
This problem asks me to use the p→q ≡~pvq and p↔q≡(~pvq) ^ (~qvp) to rewrite the given statement forms without using the symbol → or ↔.

The problem is (p→(q→r)) ↔ ((p^q)→r)

Using a bunch of truth tables and kinda just guessing I was able to come up with ~(~pv(~qvr)) v (~(p^q)vr) or (~pv(~qvr)) v ~(~(p^q)vr), both of which would work I think but I am not 100% sure they are correct so if someone wants to double check them that would be good. I think there is a quicker way to do this but the logical equivalence that I should use isn't really obvious to me just looking at a problem that complex. I guess I'm just asking for some tips on how to do this so any help would be appreciated.

You have made a minor mistake: It should be ~(~pv(~qvr)) ^ (~(p^q)vr).

De Morgan's Laws can be used to simplify logical expressions like these if required.
~(~pv(~qvr)) = (~r)^p^q and thus ~(~pv(~qvr)) ^ (~(p^q)vr) = ((~r)^(p^q))^(~(p^q) v r)