# Prove that $\neg p \to (q \to r)$ and $q \to (p \vee r)$ are logically equivalent using the laws of logical equivalences

by Nuvali   Last Updated September 12, 2019 04:20 AM

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!

Tags :

~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
May 07, 2018 16:44 PM

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.
May 07, 2018 17:00 PM