Question

Statement (P \(\Rightarrow\) Q) \(\land\) (R \(\Rightarrow\) Q) is logically equivalent to :

• A. (P \(\lor\) R) \(\Rightarrow\) Q}
• B. (P \(\Rightarrow\) R) \(\lor\) (Q \(\Rightarrow\) R)}
• C. (P \(\Rightarrow\) R) \(\land\) (Q \(\Rightarrow\) R)
• D. (P \(\land\) R) \(\Rightarrow\) Q

Answer 1

The Correct Option is A

We are given the statement (P \(\Rightarrow\) Q) \(\land\) (R \(\Rightarrow\) Q). We know that P \(\Rightarrow\) Q is equivalent to \(\neg\)P \(\lor\) Q. So, the given statement can be written as: \[ (\neg P \lor Q) \land (\neg R \lor Q). \] Using the distributive law, we can rewrite this as: \[ (\neg P \land \neg R) \lor Q. \] Using De Morgan's law, \(\neg P \land \neg R\) is equivalent to \(\neg(P \lor R)\). So the statement becomes: \[ \neg(P \lor R) \lor Q. \] This is equivalent to (P \(\lor\) R) \(\Rightarrow\) Q.