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.