Propositional Equivalence

Question

Are the following two propositions equivalent?

p IMPLIES (q IMPLIES r)

p IMPLIES (q AND r)

From what I can tell, using the logical equivalences, this should be false, correct?

p IMPLIES (q IMPLIES r)

(p IMPLIES q) IMPLIES r

This one should be true, correct?

(p IMPLIES q) AND (NOT p IMPLIES q) is equivalent to q

I believe this one is true. Is that right?

Thanks in advance!

Answer 1

The first pair are not equivalent. Take $p,r$ true, $q$ false: the first formula is true, the second is false.

The second pair are not equivalent. Take $p, r$ false, $q$ true: the first formula is true, the second is false.

The third statement is valid, yes.