Are the following logically equivalent? $\;p \rightarrow (q \rightarrow r) \text{ and }\ (p \rightarrow q) \rightarrow r$

Question

Determine whether the following pair of statements are logically equivalent or not...

$$p \rightarrow (q \rightarrow r) \;\;\text{ and }\;\; (p \rightarrow q) \rightarrow r$$

I am new to logic equations so please bare with me...

I have tried to work our the question by looking at this as an if then statement. I know the brackets have to be worked out first so i worked out $(q \rightarrow r)$ first by saying $(T \rightarrow T) \equiv T$ so $T\rightarrow T$ for the first one which is $T$.

For the second pair I again did the brackets first which turned out to be $T$ and $T \rightarrow T$ is T?

However I am unaware of is this is the right way of doing this or what $r$ means at all.

Answer 1

HINT:

Suppose $p$ is false, $q$ is true, and $r$ is false?

Answer 2

[I would like to note that the explanation below is INCORRECT. Check comments for details]

Is this necessarily the only strategy to test for equivalence between two prop statements involving implications? It seems very exhaustive to me. I've read in the comments to try analytics using method of analytic tableau but I have no contextual understanding to begin learning it.

Looking at this problem, though, I would start by recalling the definition of equivalence: those two statements are equivalent if they have the same true values for all corresponding permutations of truth values for p,q,r. I notice that they have the same form: they're both implications (albeit either the hypothesis and conclusion are also an implication). Implications are true for all permutations of truth values except for when the hypothesis is true and the conclusion is false.

Thus if these two implications were to be equivalent, they must be false for the same permutation of truth values of p,q,r that make one statement false. So, pick either one of the implications and find p,q,r that makes that implication false. Then check with the same truth values of p,q,r for the other implication to see if that is false. If they are both false for the same permutation of truth values of p,q,r they are equivalent, otherwise they aren't.

$p \rightarrow (q \rightarrow r)$ is false is false if $(q \rightarrow r)$ is false, which happens when $q$ is true and $r$ is false. Thus we need to check the case where $p$ is true, $q$ is true, $r$ is false.

You will see that the other statement will also be false given this permutation. Thus they are equivalent.