Question

Let 

p : Ramesh listens to music. 

q : Ramesh is out of his village. 

r : It is Sunday. 

s : It is Saturday. 

Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as

• A. \(\begin{array}{l} \left(\left(\sim q\right) \wedge \left(r\vee s\right)\right) \Rightarrow p \end{array}\)
• B. \( \left(q\wedge \left(r\vee s \right)\right) \Rightarrow p\)
• C. \(p\Rightarrow \left(q\wedge \left(r\vee s\right)\right)\)
• D. \(p\Rightarrow \left(\sim q \right) \wedge \left(r\vee s\right)\)

Answer 1

The Correct Option is D

To express the given statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" using logical expressions, we need to first understand the components of the statement:

• \(p\): Ramesh listens to music.
• \(q\): Ramesh is out of his village.
• \(r\): It is Sunday.
• \(s\): It is Saturday.

The phrase "Ramesh listens to music only if" indicates a conditional statement, where "only if" suggests that \(p\) is true only when a certain condition is met.

The condition for Ramesh to listen to music is that he must be not out of his village and it should be a Sunday or Saturday. In logical terms:

• Being in the village is represented as \(\sim q\).
• It is Sunday or Saturday is represented as \((r \vee s)\).
• The combined condition is \((\sim q \wedge (r \vee s))\).

Thus, the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as:

\(p \Rightarrow (\sim q \wedge (r \vee s))\)

This is the correct representation because it fully encapsulates the conditional logic described in the problem.

Now, let's verify why this option is correct and others are not:

• \((\sim q \wedge (r \vee s)) \Rightarrow p\) implies that just being in the village and it being Sunday or Saturday would cause Ramesh to listen to music, which doesn't match the "only if" structure.
• \((q \wedge (r \vee s)) \Rightarrow p\) implies being out of the village leads to listening, which is incorrect.
• \(p \Rightarrow (q \wedge (r \vee s))\) also suggests being out of the village leads to listening, conflicting with the condition.

Therefore, the correct and logically sound answer is: \(p \Rightarrow (\sim q \wedge (r \vee s))\).