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If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $F_1\wedge F_2\rightarrow F_3$ is a contradiction, then which of the following is/are necessarily true?

  1. Both $F_1$ and $F_2$ are tautologies
  2. The conjunction $F_1\wedge F_2$ is a tautology
  3. $F_3$ is a contradiction
  4. $F_1,\;F_2$ and $F_3$ all are contradictions.
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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.

 

  1. Correct
  2. Correct
  3. Correct
  4. 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.

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