Question

Verify whether the following compound propositions are tautologies or contradictions or contingency

(i) (p ∧ q) ∧ ¬ (p ∨ q) 

(ii) ((p ∨ q) ∧ ¬ p) → q

(iii) (p → q) ↔ (¬ p → q)

(iv) ((p → q) ∧ (q → r)) → (p → r)

Answer 1

(i) Truth table for (p ∧ q) ∧ ¬ (p ∨ q) 

In the above Truth table the last column entries are ‘F’. So the given propositions is a contradiction.

(ii) Truth table for ((p ∨ q) ∧ ¬ p) → q

In the above truth table the last column entries are ‘T’. So the given propositions is a tautology.

(iii) Truth table for (p → q) ↔ (¬ p → q)

In the above truth table the entries in the last column are a combination of’ T ‘ and ‘ F ‘. So the given statement is neither propositions is neither tautology nor a contradiction. It is a contingency.

(iv) Truth table for ((p → q) ∧ (q → r)) → (p → r)

The last column entires are ‘T’. So the given proposition is a tautology.