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$[(p\implies q)\implies(q\implies p)]\equiv(q \implies p)$

I believe that I have to use the rules of replacement so that I can simplify the left side of this equation. That being said, I found those rules on the internet and not in my course notes, so maybe I'm not supposed to use them. Either way, I'm a little bit stuck.

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    one way could be to turn the implications into and's and or's. Do you know how to do that ? – 2017-02-14

1 Answers 1

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Unfortunately the rules of replacement given to you do not include the following equivalence rule:

Absorption

$a \land (a \lor b) \equiv a$

$a \lor (a \land b) \equiv a$

I say 'unfortunately', because you cannot derive this equivalence from the ones that are given to you ... And you do need it to show the desired equivalence!

For here is what you get:

$(p \Rightarrow q) \Rightarrow (q \Rightarrow p) \equiv$ (Implication)

$\neg (\neg p \lor q) \lor (\neg q \lor p) \equiv$ (DeMorgan)

$(\neg \neg p \land \neg q) \lor (\neg q \lor p) \equiv$ (Double Negation)

$(p \land \neg q) \lor (\neg q \lor p) \equiv$ (Association)

$((p \land \neg q) \lor \neg q) \lor p) \equiv $ (Absorption!)

$\neg q \lor p \equiv$ (Implication)

$q \Rightarrow p$