Consider the following statements
P:Ramesh is singing
Q:Ramesh is out of his village
R:It is Sunday
S: It is Saturday
Now which of the following option is logically equivalent to the compound statement
`"Ramesh is singing only if he is in his village it is either Sunday or Saturday"`

Options

A. `((~q) wedge (r vv s)) to P`
B. `P to (~q wedge (s vv r))`
C. `P to (q wedge (r vv s))`
D. `P to (q wedge (r vv s))`

Answer

To solve the problem, we need to break down the compound statement and represent it using logical symbols.

1. **Identify the Statements**:
- Let \( P \): Ramesh is singing.
- Let \( Q \): Ramesh is out of his village.
- Let \( R \): It is Sunday.
- Let \( S \): It is Saturday.

2. **Understanding the Compound Statement**:
The statement "Ramesh is singing only if he is in his village" can be interpreted as:
- Ramesh is singing (P) implies he is not out of his village (¬Q). This can be written as:
\[ P \rightarrow \neg Q \]
This means if Ramesh is singing, then he must be in his village.

3. **Considering the Day**:
The statement also mentions "it is either Sunday or Saturday". This can be represented as:
\[ R \lor S \]
This means it is either Sunday (R) or Saturday (S).

4. **Combining the Statements**:
Now we combine both parts of the statement. The complete logical statement becomes:
\[ (P \rightarrow \neg Q) \land (R \lor S) \]
This means that Ramesh is singing only if he is in his village, and it is either Sunday or Saturday.

5. **Final Logical Expression**:
The final expression that represents the compound statement is:
\[ (P \rightarrow \neg Q) \land (R \lor S) \]

Now, we can look for an option that matches this logical expression.