If F_1, F_2 and F_3 are propositional formulae such that F_1 and F_2 - F_3 and F_1 and F_2 - not F_3 are both tautolog...

Question

If F_1, F_2 and F_3 are propositional formulae such that F_1 and F_2 -> F_3 and F_1 and F_2 -> not F_3 are both tautologies, then which of the following is true: A. Both F_1 and F_2 are tautologies   B. The conjunction F_1 and F_2 is not satisfiable   C. Neither is tautologous   D. Neither is satisfiable   E. None of the above

Accepted Answer

F1 and F2 must be unsatisfiable, so the correct option is B. Both implications are tautologies, meaning that whenever F1 and F2 is True the conclusion must be True. But the two conclusions are F3 and not F3, which can never both be True at once. The only way out is that F1 and F2 is never True at all. Hence the conjunction F1 and F2 is unsatisfiable (not satisfiable).