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 if (~p ∨ 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 "p if (~p ∨ q)" can be represented in logical terms as:
\[ p \rightarrow (\neg p \lor q) \]
Here, "→" represents implication, "¬" represents negation, and "∨" represents the logical OR.

2. **Identifying When the Implication is False**:
An implication \( A \rightarrow B \) is false only when \( A \) is true and \( B \) is false. In our case:
- \( A \) is \( p \)
- \( B \) is \( \neg p \lor q \)

Therefore, we need to find when:
- \( p \) is true
- \( \neg p \lor q \) is false

3. **Analyzing \( \neg p \lor q \)**:
The expression \( \neg p \lor q \) is false when both \( \neg p \) is false and \( q \) is false. This means:
- \( \neg p \) is false implies \( p \) is true.
- \( q \) is false.

4. **Conclusion**:
From the above analysis, we find that:
- \( p \) is true
- \( q \) is false

Thus, the truth values of \( p \) and \( q \) are:
- \( p = \text{True} \)
- \( q = \text{False} \)

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

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