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If $F_1$, $F_2$ and $F_3$ are propositional formulae such that $F_1 \land F_2 \rightarrow F_3$ and $F_1 \land F_2 \rightarrow \sim F_3$ are both tautologies, then which of the following is true:

- Both $F_1$ and $F_2$ are tautologies
- The conjunction $F_1 \land F_2$ is not satisfiable
- Neither is tautologous
- Neither is satisfiable
- None of the above

### 4 Comments

## 5 Answers

### 13 Comments

edited
Jun 14, 2017
by rahul sharma 5

How is RHS is always true?

Can i re-write as:-

f1 ^ f2 ==> f3 ^ ~f3,here it will become f1 ^ f2 ==>0,

or

f1 ^ f2 ==> f3 or ~f3,here it will become f1 ^ f2 ==>1,means lhs should be 0 if rhs is always true.

Which version is correct?Or is their some other approach we follow?If this this wrong approach then please tell why?

Can i re-write as:-

f1 ^ f2 ==> f3 ^ ~f3,here it will become f1 ^ f2 ==>0,

or

f1 ^ f2 ==> f3 or ~f3,here it will become f1 ^ f2 ==>1,means lhs should be 0 if rhs is always true.

Which version is correct?Or is their some other approach we follow?If this this wrong approach then please tell why?

" F1∧F2→F3 and F1∧F2→∼F3 are both tautologies " it is possible in 2 cases

case 1) True→True

case 2) False→False/True

here F3 is in both F3 and ∼F3 form so only case 2) will apply

so F1∧F2 is False means F1=False and F2=False

(a). "Both F1 and F2 are tautologies" is INCORRECT

(b). "The conjunction F1∧F2 is not satisfiable" is INCORRECT becoz for being satisfiable atleast one possibility of F1 and F2 should be True

(c). "Neither is tautologous" is INCORRECT as both are Tautologies

(d). "Neither is satisfiable " is INCOORECT as both are Tautologies so also satisfiable

Hence correct ans is B

case 1) True→True

case 2) False→False/True

here F3 is in both F3 and ∼F3 form so only case 2) will apply

so F1∧F2 is False means F1=False and F2=False

(a). "Both F1 and F2 are tautologies" is INCORRECT

(b). "The conjunction F1∧F2 is not satisfiable" is INCORRECT becoz for being satisfiable atleast one possibility of F1 and F2 should be True

(c). "Neither is tautologous" is INCORRECT as both are Tautologies

(d). "Neither is satisfiable " is INCOORECT as both are Tautologies so also satisfiable

Hence correct ans is B

### 7 Comments

F1∧F2→F3

F1∧F2→∼F3

Now take **contrapositive** of both

∼F3→∼(F1∧F2)

F3→∼(F1∧F2)

--------------------------------------------by using **resolution rule**

∼(F1∧F2) or ∼(F1∧F2)=∼(F1∧F2)

which means **Nand** of f1 and f2 is tautology ,therefore** conjuction of both of them is contradiction** hence not satisfiable.

so optio**n B i**s th answer.

Let P1: F1∧F2→F3 and P2: F1∧F2→∼F3 is two premises ,so we need to findout what is the conclusion .

P1: F1∧F2→F3

P2: F1∧F2→∼F3

Both the premises will simultaneously true when (F1∧F2) is False i.e ~(F1∧F2).

Hence the conclusion is ~(F1∧F2) i.e the conjunction (F1∧F2)is not satisfiable.

So ans is (b). The conjunction F1∧F2 is not satisfiable