Key Concepts and Formulas

• "p only if q": This logical statement is equivalent to "if p, then q", which is represented as $p \Rightarrow q$.
• Negation: The negation of a statement 'p' is denoted as $\sim p$.
• Conjunction: The conjunction of two statements 'p' and 'q' is denoted as $p \wedge q$ (p and q).
• Disjunction: The disjunction of two statements 'p' and 'q' is denoted as $p \vee q$ (p or q).

Step-by-Step Solution

Step 1: Translate the given statements into symbolic form.

We are given:

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

Step 2: Express "Ramesh is in his village" in symbolic form.

Since q represents "Ramesh is out of his village", the statement "Ramesh is in his village" is the negation of q, which is represented as $\sim q$.

Step 3: Express "It is Sunday or Saturday" in symbolic form.

This is the disjunction of r and s, which is represented as $r \vee s$.

Step 4: Express "Ramesh is in his village and it is Sunday or Saturday" in symbolic form.

This is the conjunction of "Ramesh is in his village" (which is $\sim q$) and "It is Sunday or Saturday" (which is $r \vee s$). Therefore, the combined statement is $(\sim q) \wedge (r \vee s)$.

Step 5: Express "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" in symbolic form.

The phrase "p only if q" is equivalent to "if p, then q", which is represented as $p \Rightarrow q$. In our case, p is "Ramesh listens to music" and q is "Ramesh is in his village and it is Sunday or Saturday". Therefore, the given statement can be written as: $p \Rightarrow ((\sim q) \wedge (r \vee s))$

Step 6: Compare the result with the given options.

The derived expression, $p \Rightarrow ((\sim q) \wedge (r \vee s))$, corresponds to option (D).

Common Mistakes & Tips

• Pay close attention to the "only if" construct. "p only if q" means that p can only happen if q also happens. This translates to $p \Rightarrow q$. It does not translate to $q \Rightarrow p$.
• Understand the difference between conjunction (and) and disjunction (or). Conjunction requires both statements to be true, while disjunction requires at least one of the statements to be true.
• Carefully negate statements. The negation of "Ramesh is out of his village" is "Ramesh is not out of his village," which means "Ramesh is in his village."

Summary

The given statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" is translated into symbolic form by first representing each simple statement (p, q, r, s) with its corresponding variable. Then, "Ramesh is in his village" is expressed as $\sim q$, and "It is Sunday or Saturday" is expressed as $r \vee s$. The phrase "Ramesh is in his village and it is Sunday or Saturday" is then expressed as $(\sim q) \wedge (r \vee s)$. Finally, "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" is expressed as $p \Rightarrow ((\sim q) \wedge (r \vee s))$.

The final answer is \boxed{D}.