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Math Help - Help with logical equivalences

  1. #1
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    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.
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  2. #2
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    Try working from the inside out. That is, start with

    (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?
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  3. #3
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    Quote Originally Posted by learningguy View Post
    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)
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