Propositional Logic: Confused on using operators with equal precedence

Question

My problem involves using operators of equal precedence in propositional logic.

I'm trying to use truth tables to prove the following:

p ⇔ q ⇔ (p ⇒ q) ∧ (q ⇒ p)

My problem however stems from the fact that I can see 2 ways to go about this. One being:

(p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p))

and the other:

p ⇔ (q ⇔ (p ⇒ q) ∧ (q ⇒ p))

When writing out the truth tables for each, both of them result in a tautology, as can be seen here:

Truth table for (p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p))

Truth table for p ⇔ (q ⇔ (p ⇒ q) ∧ (q ⇒ p))

I'm certain that only one of those is correct. Would someone be able to explain which one and why?

Answer 1

Your second way is the right way according to reference here:

When an operand is surrounded by operators of equal precedence, the operand associates to the right.

p ⇒ q ⇒ r: (p ⇒ (q ⇒ r))

p ⇒ q ⇔ r: (p ⇒ (q ⇔ r))