If the statements `(p^^~r) to (qvvr)`, q and r are all false, then p

Options

A. is true
B. is false
C. may be true or false
D. data is insufficient

Explanation

To solve the problem, we need to analyze the logical statements given and determine the truth value of \( p \) based on the information that \( q \) and \( r \) are both false.

### Step-by-step Solution:

1. **Identify the Given Statements:**
We have three statements:
- \( p \land \neg r \to (q \lor r) \)
- \( q \) (which is false)
- \( r \) (which is false)

2. **Substitute the Values of \( q \) and \( r \):**
Since \( q \) and \( r \) are both false, we can substitute these values into the first statement:
- \( p \land \neg r \to (q \lor r) \)
- This becomes \( p \land \neg \text{false} \to (\text{false} \lor \text{false}) \)
- Simplifying further, we have \( p \land \text{true} \to \text{false} \)
- This simplifies to \( p \to \text{false} \)

3. **Analyze the Implication:**
The implication \( p \to \text{false} \) is only false when \( p \) is true. Therefore, for the entire statement to be false, \( p \) must be true.

4. **Conclusion:**
Since \( p \) must be true for the implication to hold, we conclude:
- \( p = \text{true} \)

### Final Answer:
Thus, the value of \( p \) is **true**.