Which of the following is/are tautology?

To determine which expressions are tautologies, we evaluate each option for all possible truth values of the propositional variables.

Option A: p ∨ ¬p

This is the law of excluded middle. It is true when p is true and also when p is false. Therefore, it is always true and is a tautology.

Option B: p ∧ p ∨ q

This simplifies to p ∨ q. It is false when both p and q are false. Therefore, it is not a tautology.

Option C: q → p ∧ p → q

This is equivalent to (q → p) ∧ (p → q), which is the biconditional p ↔ q. It is true only when p and q have the same truth value. Therefore, it is not a tautology.

Option D: p → (q → (r → ¬p))

This expression is false when p is true, q is true, r is true, and ¬p is false. In this case, the inner implication (r → ¬p) is false, making the entire expression false. Therefore, it is not a tautology.

Thus, only Option A is a tautology.