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3 votes
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In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that

$$\begin{array}{c} (P \to Q) \wedge (R \to S) \\  P \vee R \\ \hline Y \\ \hline \end{array}$$

$Y$ is the premise :

  1. $P \vee R$
  2. $P \vee S$
  3. $Q \vee R$
  4. $Q \vee S$
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6 Answers

Best answer
1 votes
1 votes

Answer: A, B,C, D

Each option is a Valid Conclusion of the given premises.

  1. $P \vee R$ trivially follows from $P \vee R.$
  1. $P \vee R$ is given. If $R$ then due to $R \rightarrow S,$ $S$ will follow, hence, $P \vee S$ will follow. If $P$ then $P \vee S$ will follow. So, in every case, $P \vee S$ will follow.
  1. $P \vee R$ is given. If $P$ then due to $P \rightarrow Q,$ $Q$ will follow, hence, $Q \vee R$ will follow. If $R$ then $Q \vee R$ will follow. So, in every case, $Q \vee R$ will follow.
  1. $P \vee R$ is given. If $P$ then due to $P \rightarrow Q,$ $Q$ will follow, hence, $Q \vee S$ will follow. If $R$ then due to $R \rightarrow S,$ $S$ will follow, hence, $Q \vee S$ will follow.

A nice question, But the answer will be All the options.

edited by
4 votes
4 votes

Given that premises are

 (P→Q)˄(R→S)

 (P˅R) 

   (P→Q)   = ~PVQ

   (R→S)  = ~RVS

   (P˅R) 


  Q V S

There will be Resolution (rule of inference ) between these premises to give conclusion  

~ P & P ,  R & R' will resolve out and then we  construct the disjunction of the remaining clauses

  to give SVQ option 4)

2 votes
2 votes

One can think like -

For P-->Q: If you work hard(P) then you will qualify NET(Q).

For R-->S: If you play game(R) then you will get gold(S).

Given premise is PVR : means if u work hard(P) OR(v) if you play game(R), then conclusion will be-

than You will qualify NET(Q) OR(v) you will get gold(S). Which is nothing but QVS.

Hence, Option D is correct.

1 votes
1 votes

$(P\to Q)\wedge (R\to S)$  means if $P$ is true, then $Q$ has to be true, and if $R$ is true then $S$ has to be true.

The second statement $P \vee R$, says that either $P$ or $R$ is true, which means either $Q$ or $S$ should be true. 

Correct option is (D)  $Q\vee S$.

Answer:

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