Exercise 1.6 (4)
4. Prove that each of the following statement pattern is a contradiction.
i) (p ∨ q) ∧ (∼ p ∧ ∼ q)
Ans:
p | q | ∼ p | ∼ q | p ∨ q | ∼ p ∧ ∼ q | (p ∨ q) ∧ (∼ p ∧ ∼ q) |
T | T | F | F | T | F | F |
T | F | F | T | T | F | F |
F | T | T | F | T | F | F |
F | F | T | T | F | T | F |
All the truth value in the last column are F. Hence, it is contradiction
ii) (p ∧ q) ∧ ∼ p
Ans:
p | q | ∼ p | p ∧ q | (p ∧ q) ∧ ∼ p |
T | T | F | T | F |
T | F | F | F | F |
F | T | T | F | F |
F | F | T | F | F |
All the truth value in the last column are F. Hence, it is contradiction
iii) (p ∧ q) ∧ (∼ p ∨ ∼ q)
Ans:
p | q | ∼ p | ∼ q | p ∧ q | ∼ p ∨ ∼ q | (p ∧ q) ∧ (∼ p ∨ ∼ q) |
T | T | F | F | T | F | F |
T | F | F | T | F | T | F |
F | T | T | F | F | T | F |
F | F | T | T | F | T | F |
All the truth value in the last column are F. Hence, it is contradiction
iv) (p → q) ∧ (p ∧ ∼ q)
Ans:
p | q | ∼ q | p → q | p ∧ ∼ q | (p → q) ∧ (p ∧ ∼ q) |
T | T | F | T | F | F |
T | F | T | F | T | F |
F | T | F | T | F | F |
F | F | T | T | F | F |
All the truth value in the last column are F. Hence, it is contradiction