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Let $P, Q, R$ and $S$ be Propositions. Assume that the equivalences $P \Leftrightarrow (Q \vee \neg Q )$ and $Q \Leftrightarrow R$ hold. Then the truth value of the formula $(P \wedge Q) \Rightarrow ((P \wedge R) \vee S)$ is always

  1. True
  2. False
  3. Same as truth table of $Q$
  4. Same as truth table of $S$
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1 Answer

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P ⇔ (Q ∨ ¬ Q) "P should be true because RHS will be TRUE always "

Q ⇔ R "when Q is true R is true" and  "when Q is false R is false"

 $(P ∧ Q) ⇒ ((P ∧ R) ∨ S)$

there can be only 2 cases (value of S doesn't matter)

1) P = True, Q = True and R = True

        $(T ∧ T) ⇒ ((T ∧ T) ∨ S)$ 

         so this case is True

2) P =True, Q = R = False

        $(T ∧ F) ⇒ ((P ∧ R) ∨ S)$

        In case implication if the premises is false then whole statement is true

        this case is also true

The given expression is True in both the cases.

Answer is 1) True

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What if R is false in your ist point ?
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