Given the propositional statement: (~q ⋀ (p→q))→~p, which of the following…

To determine the nature of the propositional statement (~q ⋀ (p→q))→~p, we simplify it using logical equivalences.

Step 1: Rewrite the implication p→q as ~p ⋁ q.

So, (~q ⋀ (p→q))→~p becomes (~q ⋀ (~p ⋁ q))→~p.

Step 2: Apply distributive law: ~q ⋀ (~p ⋁ q) ≡ (~q ⋀ ~p) ⋁ (~q ⋀ q).

Since ~q ⋀ q is a contradiction (always false), the expression simplifies to (~q ⋀ ~p).

So, the original statement becomes (~q ⋀ ~p)→~p.

Step 3: Rewrite the implication: (~q ⋀ ~p)→~p ≡ ~(~q ⋀ ~p) ⋁ ~p.

Apply De Morgan's law: ~(~q ⋀ ~p) ≡ q ⋁ p.

So, the expression becomes (q ⋁ p) ⋁ ~p.

Step 4: Simplify using associativity and complement law: p ⋁ ~p ≡ 1.

Thus, (q ⋁ p) ⋁ ~p ≡ q ⋁ (p ⋁ ~p) ≡ q ⋁ 1 ≡ 1.

Since the expression simplifies to 1 (true) for all truth values of p and q, it is a tautology.