# Exercise 1.6 (6)

6. Using the truth table, verify
i) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
Ans:

 1 2 3 4 5 6 7 8 p q r q ∧ r p ∨ (q ∧ r) (p ∨ q) (p ∨ r) (p ∨ q) ∧ (p ∨ r) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

The entries in the Columns 5 and 8 are identical.
∴ p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

ii) p → (p → q) ≡ ∼ q → (p → q)
Ans:

 1 2 3 4 5 6 p q ~q p → q p → (p → q) ∼ q → (p → q) T T F T T T T F T F F F F T F T T T F F T T T T

In the above truth table, entries in columns 5 and 6 are identical.
∴ p → (p → q) ≡ ∼ q → (p → q)

iii) ∼ (p → ∼ q) ≡ p ∧ ∼ (∼ q) ≡ p ∧ q
Ans:

 1 2 3 4 5 6 7 8 p q ~q p → ∼ q ∼ (p → ∼ q) ∼ (∼ q) p ∧ ∼ (∼ q) p ∧ q T T F F T T T T T F T T F F F F F T F T F T F F F F T T F F F F

In the above table, entries in the columns 5, 7 & 8 are identical.
∴∼ (p → ∼ q) ≡ p ∧ ∼ (∼ q) ≡ p ∧ q

iv) ∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p
Ans:

 1 2 3 4 5 6 7 P q ~p (p ∨ q) ∼ (p ∨ q) (∼ p ∧ q) ∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p T T F T F F F T F F T F F F F T T T F T T F F T F T F T

In the above truth table, the entries in the Columns 3 and 7 are identical.
∴ ∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p