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