Unlike the binary connectives {∧,∨, ⊕,⇔}, implication is not associative. In other words, p ⇒ (q ⇒ r) and (p ⇒ q) ⇒ r are not logically equivalent. The next few exercises explore the non-associativity...

Unlike the binary connectives {∧,∨, ⊕,⇔}, implication is not associative. In other words, p ⇒ (q ⇒ r) and (p ⇒ q) ⇒ r are not logically equivalent. The next few exercises explore the non-associativity of ⇒.

1. Prove that implication is not associative by giving a truth assignment in which p ⇒ (q ⇒ r) and (p ⇒ q) ⇒ r have different truth values.

2. Consider the propositions p ⇒ (q ⇒ q) and (p ⇒ q) ⇒ q. One of these is a tautology; one of them is not. Which is which? Prove your answer.

3.Consider the propositions p ⇒ (p ⇒ q) and (p ⇒ p) ⇒ q. Is either one a tautology? Satisfiable? Unsatisfiable? What is the simplest proposition to which each is logically equivalent?