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