Solution
The correct answer is option 4.
Key Points
Option A: ((p → r) ∧ (q → r)) and ((p ∨ q) → r)
Both are equal and it gives the same truth table. Hence it is logically equivalent.
Option B: p ↔ q and (¬ p ↔ ¬ q)
(p → q)(q → p) and (¬ p → q)(¬ q →p)
(p+q̅)(q+p̅) and (p+q̅)(q+p̅)
Both are equal and it gives the same truth table. Hence it is logically equivalent.
Option C: (p → q) ∧ (q → p) and p ↔ q
Both tables give equal values. Hence it is logically equivalent.
Option D: ((p ∧ q) → r ) and ((p → r) ∧ (q → r))
| P |
Q |
R |
¬P |
¬Q |
P⇒R |
Q⇒R |
(P⇒R)∧(Q⇒R) |
P∧Q |
P∧Q⇒R |
| T |
T |
T |
F |
F |
T |
T |
T |
T |
T |
| T |
T |
F |
F |
F |
F |
F |
F |
T |
F |
| T |
F |
T |
F |
T |
T |
T |
T |
F |
T |
| T |
F |
F |
F |
T |
F |
T |
F |
F |
T |
| F |
T |
T |
T |
F |
T |
T |
T |
F |
T |
| F |
T |
F |
T |
F |
T |
F |
F |
F |
T |
| F |
F |
T |
T |
T |
T |
T |
T |
F |
T |
| F |
F |
F |
T |
T |
T |
T |
T |
F |
T |
The above truth table is not equivalent. Hence the above statement is True, Logically not equivalent.
∴ Hence the correct answer is ((p ∧ q) → r ) and ((p → r) ∧ (q → r)).
