How exactly does the placement of parenthesis change a material conditional in propositional logic?

Question

I noticed that with exportation P → (Q → R) is equivalent to (P ∧ Q) → R. This got me confused about the difference between P→ (Q → R) and (P → Q) → R. What exactly do the parentheses do to change the meaning or are they equivalent?

Answer 1

You could make a truth table. Note that if $Q$ is false, $P\rightarrow (Q \rightarrow R)$ is always true, because the consequent is true. On the other hand $(P \rightarrow Q) \rightarrow R$ is false if $P,Q,R$ are all false because the antecedent is true and the consequent is false.

Answer 2

Contraposition law says that: $(X \rightarrow Y) \equiv ( \neg Y \rightarrow\neg X)$ .

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$P → (Q → R) \equiv \neg (Q\rightarrow R) \rightarrow\neg P\equiv (Q\land\neg R)\rightarrow\neg P$

So, it means that $\neg R$ conjoined with $Q$ is sufficient for $P$ to be false.

But

$(P\rightarrow Q) \rightarrow R \equiv \neg R \rightarrow \neg ( P \rightarrow Q)\equiv \neg R \rightarrow (P\land \neg Q)$

So it means that $\neg R$ is sufficient for $P$ to be true.