Which of the following are tautology?
(A) (P → (P ∧ Q)) → (P → Q)
(B) ((P → Q) → Q) → (P ∨ Q)
(C) ((P ∧ ¬ P) → Q) → ((P ∨ ¬ P) → R)
(D) (Q → (P ∧ ¬ P)) → (R → (P ∧ ¬ P))
Choose the correct answer from the options given below:
Solution
The correct answer is (A) and (B) Only
SOLUTION:
(A) (P → (P ∧ Q)) → (P → Q)
= (¬ P v (P ∧ Q)) → (P → Q)
= (¬ P v P ∧ ¬ P v Q) → (P → Q)
= (¬ P v Q) → (P → Q)
= (¬ P v Q) → (¬ P v Q)
= ¬ (¬ P v Q) v (¬ P v Q)
= ¬ ¬ P ∧ ¬ Q v (¬ P v Q)
= P ∧ ¬ Q v (¬ P v Q)
= P v ¬ P ∧ ¬ Q v ¬ P v Q = 1
(B) ((P → Q) → Q) → (P ∨ Q)
= ((¬ P v Q) → Q) → (P ∨ Q)
= (¬ (¬ P v Q) v Q) → (P ∨ Q)
= (P ∧ ¬ Q) v Q) → (P ∨ Q)
= (P v Q ∧ ¬ Q v Q) → (P ∨ Q)
= (P v Q) → (P ∨ Q)
= ¬ (P v Q) v (P v Q)
= ¬ P ∧ ¬ Q v (P v Q)
= ¬ P v P ∧ ¬ Q v Q = 1
So, above solution both A and B are tautology.