@Deepak Poonia Thank you Sir. I understood my mistake.

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If $F1$, and $F2$ are propositional formulae/expressions, over same set of propositional variables, such that $F1,F2$ both are contingencies, then which of the following is/are necessarily false(i.e. Never Possible):

- $F1 \vee F2$ is a contingency.
- $F1 \vee F2$ is a tautology.
- $F1 \vee F2$ is a contradiction
- $(F1 \rightarrow F2) \vee (F2 \rightarrow F1)$ is contingency.

4 votes

**Answer: C, D**

If $F1$ and $F2$ are contingencies.

- This is obvious that it is not necessarily false since both $F1$ and $F2$ are contingency, $F1 \lor F2$ is a contingency.

- $F1 \lor F2$ is tautology is not necessarily false because conside the situation where either of $F1$ or $F2$ is true when other is false. This will give $F1 \lor F2$ a tautology.
- $F1 \lor F2$ is a contradiction is necessarily false because to be a contraction $F1 \lor F2$ must always be false for that both $F1$ and $F2$ must be contradiction.
- $(F1→F2)∨(F2→F1)$ = $\lnot F1 \lor F2 \lor \lnot F2 \lor F1$ is always tautology regardless what logical value $F1$ and $F2$ takes, so option (D) is also necessarily false.

Correction in statement A:

- This is obvious that it is not necessarily false since both F1 and F2 are contingency, F1∨F2 is a contingency.

Since $F_1, F_2$ both are contingency, So, $F_1 \vee F_2$ may** be** contingency, but it is Not necessary that $F_1 \vee F_2$ will definitely be contingency. For example, Let $F_1 = P, F_2 = \neg P,$ here, $F_1 \vee F_2 $ is Tautology, not contingency.

Correction in statement B:

- F1∨F2 is tautology is not necessarily false because conside the situation where either of F1 or F2 is true when other is false. This will give F1∨F2 a tautology.

Since $F_1, F_2$ both are contingency, So, $F_1 \vee F_2$ may** be** Tautology, but it is Not necessary that $F_1 \vee F_2$ will definitely be a Tautology. For example, Let $F_1 = P, F_2 = \neg P,$ here, $F_1 \vee F_2 $ is Tautology, but if $F_1 = P, F_2 = P,$ here, $F_1 \vee F_2 $ is Not Tautology.

Another example, $F_1 = P\vee Q, F_2 = P \rightarrow Q,$ here, $F_1 \vee F_2 $ is Tautology

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