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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):

  1. $F1 \vee F2$ is a contingency.
  2. $F1 \vee F2$ is a tautology.
  3. $F1 \vee F2$ is a contradiction
  4. $(F1 \rightarrow F2) \vee (F2 \rightarrow F1)$ is contingency.
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Answer: C, D

If $F1$ and $F2$ are contingencies.

  1. This is obvious that it is not necessarily false since both $F1$ and $F2$ are contingency, $F1 \lor F2$ is a contingency.
  1. $F1 \lor F2$ 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.
  2. $F1 \lor F2$ 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.
  3. $(F1→F2)∨(F2→F1)$ = $\lnot F1 \lor F2 \lor \lnot F2 \lor F1$ is always tautology regardless what logical value $F1$ and $F2$ takes, so option (D) is also necessarily false.

 

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