# Exercise 1.6 (7)

7. Prove that the following pairs of statement patterns are equivalent.
i) p ∨ (q ∧ r) and (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 ↔ q and (p → q) ∧ (q → p)
Ans:

 1 2 3 4 5 6 p q p ↔ q p → q q → p (p → q) ∧ (q → p) T T T T T T T F F F T F F T F T F F F F T T T T

In the above table, entries in columns 3 and 6 are identical.
∴Statement p ↔ q and (p → q) ∧ (q → p) are equivalent.

iii) p → q and ∼ q → ∼ p and ∼ p ∨ q
Ans:

 1 2 3 4 5 6 7 p q ∼ p ∼ q p → q ∼ q → ∼ p ∼ p ∨ q T T F F T T T T F F T F F F F T T F T T T F F T T T T T

In the above table, entries in columns 5, 6 and 7 are identical.
∴Statement p → q and ∼ q → ∼ p and ∼ p ∨ q are equivalent.

iv) ∼ (p ∧ q) and ∼ p ∨ ∼ q.
Ans:

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

In the above table, entries in column 6 and 7 are identical.
∴ Statement ∼ (p ∧ q) and ∼ p ∨ ∼ q are equivalent.