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Please help, I cannot figure out how $\neg p \to (q \to r)$ and $q \to (p \vee r)$ are logically equivalent using the laws of logical equivalences.

Here's what I came up, please help to explain how to show that the equation is logically equivalent using the laws of logical equivalences

$\neg p \to (q \to r) = q \to (p \vee r)$

$\neg p (\neg q \vee r) = q (p \vee r)$

$p (q \vee r) = q \to p$

Thank you!

Peter Kagey
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Nuvali
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    use $p \rightarrow q = \neg p \lor q$ .. you seem to change $p \rightarrow q$ into $\neg p \ q$ ... which I interpret as $\neg p \land q$ ... but it should be $\neg p \color{red}\lor q$! So, for example, $q \rightarrow (p \lor r) = \neg q \lor (p \lor r)$. ... Also, I have no idea what you do on that last line, but that's certainly not correct – Bram28 May 07 '18 at 16:35

2 Answers2

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~p --> (q --> r)
~~p v (q --> r)
p v (~q v r)
p v (~q) v r
~q v (p v r)
q --> (p v r)

Юрій Ярош
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We have $(\neg p\to (q\to r))\equiv (p\lor (q\to r))\equiv (p\lor\neg q\lor r)\equiv (\neg q \lor p\lor r)\equiv (q\to (p\lor r))$.

J.G.
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