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The first order logic (FOL) statement $((R\vee Q)\wedge(P\vee \neg Q))$ is equivalent to which of the following?

  1. $((R\vee \neg Q)\wedge(P\vee \neg Q)\wedge (R\vee P))$
  2. $((R\vee Q)\wedge(P\vee \neg Q)\wedge (R\vee P))$
  3. $((R\vee Q)\wedge(P\vee \neg Q)\wedge(R\vee \neg P))$
  4. $((R\vee Q)\wedge(P\vee \neg Q)\wedge (\neg R\vee P))$
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Consider $G=((R \vee Q)\wedge(P \vee \sim Q))$

$\equiv (P\wedge R)\vee(\sim Q \wedge R)\vee(P \wedge Q)\vee(Q \wedge \sim Q)$

$\equiv (P\wedge R)\vee(\sim Q \wedge R)\vee(P \wedge Q)\vee F$

$\equiv (P\wedge R)\vee(\sim Q \wedge R)\vee(P \wedge Q)$

 

Now

$B . ((R \vee Q) \wedge (P \vee \lnot Q) \wedge (R \vee P))$

$\equiv ((R \vee Q) \wedge ((P \wedge R) \vee(P \wedge P) \vee (\sim Q \wedge R) \vee(P \wedge \sim Q))$

$\equiv ((R \vee Q) \wedge ((P \wedge R) \vee P \vee (\sim Q \wedge R) \vee(P \wedge \sim Q))$

$\equiv ((P \wedge R ) \vee (P \wedge R) \vee (\sim Q \wedge R ) \vee(P \wedge \sim Q \wedge R) \vee(P \wedge Q \wedge R)\vee(P \wedge Q)\vee(Q\wedge \sim Q \wedge R)\vee (P \wedge \sim Q \wedge Q)$

$\equiv ((P \wedge R ) \vee (P \wedge R) \vee (\sim Q \wedge R ) \vee(P \wedge \sim Q \wedge R) \vee(P \wedge Q \wedge R)\vee(P \wedge Q)\vee(F \wedge R)\vee (P \wedge F)$

$\equiv ((P \wedge R ) \vee (P \wedge R) \vee (\sim Q \wedge R ) \vee(P \wedge \sim Q \wedge R) \vee(P \wedge Q \wedge R)\vee(P \wedge Q)\vee F\vee F$

$\equiv ( (P \wedge R) \vee(P \wedge Q \wedge R) (\sim Q \wedge R ) \vee(P \wedge \sim Q \wedge R)\vee(P \wedge Q)$

$\equiv ( (P \wedge R) \vee (\sim Q \wedge R ) \vee(P \wedge Q)$

$\equiv G$

 

Hence,Option$(B) ((R \vee Q) \wedge (P \vee \lnot Q) \wedge (R \vee P))$.
Answer:

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