F1 ^ F2 → F3
Implication statement returns False only when the condition is True and conclusion is False.
Since F1 ^ F2 → F3 is a contradiction, this means that F1 ^ F2 = True and F3 = False.
For F1 ^ F2 to be True, F1 must be True and F2 must be True.
This exactly means that F1, F2 are both tautologies and F3 is a contradiction.
- Correct
- Correct
- Correct
- Wrong, since F1 and F2 are tautologies.
Alternative solution (Time-Taking),
We may consider using a truth table and eliminating the possibilities that would never occur for the given conditions in the problem.
F1 |
F2 |
F3 |
F1 ^ F2 |
F1 ^ F2 → F3 |
F |
F |
F |
F |
T |
F |
F |
T |
F |
T |
F |
T |
F |
F |
T |
F |
T |
T |
F |
T |
T |
F |
F |
F |
T |
T |
F |
T |
F |
T |
T |
T |
F |
T |
F |
T |
T |
T |
T |
T |
F1 ^ F2 → F3 is contradiction if and only if all the rows below it are False (F) in the truth table.
Therefore given the formulae F1 ^ F2 → F3 is contradiction, the possibilities reduce to –
F1 |
F2 |
F3 |
F1 ^ F2 |
F1 ^ F2 → F3 |
T |
T |
F |
T |
F |
Clearly, F1 is tautology, F2 is tautology and F3 is contradiction.