The compound statement `p to ( ~ p ^^ q)` is false, then the truth values of p and q are respectively.

Options

A. T,T
B. T,F
C. F,T
D. F,F

Explanation

To solve the problem, we need to analyze the compound statement \( p \to (\neg p \land q) \) and determine under what conditions it is false.

### Step-by-Step Solution:

1. **Understanding the Conditional Statement**:
The statement \( p \to (\neg p \land q) \) is a conditional statement, which is false only when the antecedent \( p \) is true and the consequent \( \neg p \land q \) is false.

2. **Identifying When the Consequent is False**:
The expression \( \neg p \land q \) is false if either \( \neg p \) is false or \( q \) is false. Since \( \neg p \) is the negation of \( p \), we can conclude:
- If \( p \) is true, then \( \neg p \) is false.
- Therefore, for \( \neg p \land q \) to be false, \( q \) must also be false.

3. **Setting the Values**:
- From the above analysis, we have:
- \( p = \text{True} \)
- \( q = \text{False} \)

4. **Verifying the Values**:
- If \( p = \text{True} \), then \( \neg p = \text{False} \).
- Thus, \( \neg p \land q = \text{False} \land q \).
- Since \( q \) is false, \( \neg p \land q \) is indeed false.
- Therefore, the entire statement \( p \to (\neg p \land q) \) becomes \( \text{True} \to \text{False} \), which is false.

5. **Considering Other Possible Values for \( q \)**:
- If \( p = \text{True} \) and \( q = \text{True} \), then \( \neg p \land q = \text{False} \land \text{True} = \text{False} \).
- So, \( p \to (\neg p \land q) \) would still be \( \text{True} \to \text{False} \), which is also false.

### Conclusion:
The truth values of \( p \) and \( q \) that make the compound statement false are:
- \( p = \text{True} \)
- \( q = \text{True} \) or \( q = \text{False} \)

Thus, the possible pairs of truth values for \( (p, q) \) are:
- (True, True)
- (True, False)