If `p to (~ p vvq)` is false, the truth values of p and q are , respectively

Options

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

Explanation

To solve the problem, we need to analyze the statement \( p \to (\neg p \lor q) \) and determine the truth values of \( p \) and \( q \) when this statement is false.

### Step-by-Step Solution:

1. **Understanding the Statement**:
The statement we are analyzing is \( p \to (\neg p \lor q) \). The implication \( p \to A \) is false only when \( p \) is true and \( A \) is false.

2. **Negating \( p \)**:
We need to find \( \neg p \) (the negation of \( p \)). If \( p \) is true (T), then \( \neg p \) is false (F), and if \( p \) is false (F), then \( \neg p \) is true (T).

3. **Constructing a Truth Table**:
We will create a truth table for \( p \) and \( q \) to evaluate the expression \( \neg p \lor q \).

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \lor q \) | \( p \to (\neg p \lor q) \) |
|---------|---------|---------------|----------------------|------------------------------|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

4. **Finding When the Statement is False**:
From the truth table, we see that \( p \to (\neg p \lor q) \) is false only in the case where \( p \) is true (T) and \( q \) is false (F).

5. **Conclusion**:
Therefore, the truth values of \( p \) and \( q \) that make the statement \( p \to (\neg p \lor q) \) false are:
- \( p = T \)
- \( q = F \)

### Final Answer:
The truth values of \( p \) and \( q \) are \( T \) (True) and \( F \) (False), respectively.

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