Exercise 1.6 (3)
3. Prove that each of the following statement pattern is a tautology.
i) (p ∧ q) → q
Ans:
p | q | p ∧ q | (p ∧ q) → q |
T | T | T | T |
T | F | F | T |
F | T | F | T |
F | F | F | T |
All the truth value in the last column are T. Hence , it is tautology.
ii) (p → q) ↔ (∼ q → ∼ p)
Ans:
p | q | ∼ p | ∼ q | p → q | ∼ q → ∼ p | (p → q) ↔ (∼ q → ∼ p) |
T | T | F | F | T | T | T |
T | F | F | T | F | F | T |
F | T | T | F | T | T | T |
F | F | T | T | T | T | T |
All the truth value in the last column are T. Hence , it is tautology.
iii) (∼ p ∧ ∼ q) → (p → q)
Ans:
p | q | ∼ p | ∼ q | ∼ p ∧ ∼ q | p → q | (∼ p ∧ ∼ q) → (p → q) |
T | T | F | F | F | T | T |
T | F | F | T | F | F | T |
F | T | T | F | F | T | T |
F | F | T | T | T | T | T |
All the truth value in the last column are T. Hence , it is tautology.
iv) (∼ p ∨ ∼ q) ↔ ∼ (p ∧ q)
Ans:
p | q | ∼ p | ∼ q | ∼ p ∨ ∼ q | p ∧ q | ∼ (p ∧ q) | (∼ p ∨ ∼ q) ↔ ∼ (p ∧ q) |
T | T | F | F | F | T | F | T |
T | F | F | T | T | F | T | T |
F | T | T | F | T | F | T | T |
F | F | T | T | T | F | T | T |
All the truth value in the last column are T. Hence , it is tautology.