Which of the following are tautology ? (A) (P → (P ∧ Q)) → (P → Q) (B) ((P → Q) → Q) → (P ∨ Q) (C) ((P…

In A, $P \to (P \wedge Q)$ is equivalent to $P \to Q$, so A has the form X → X and is a tautology. In B, $(P \to Q) \to Q$ is equivalent to $P \vee Q$, so B also has the form X → X. Since P ∨ ¬P is always true, C reduces to Q → R, false when Q is true and R false, and D reduces to ¬Q → ¬R, that is R → Q, false when R is true and Q false. So only A and B are tautologies.