If `(p^^~r) to (~p vvq)` is false, then truth values of p,q and r are respectively.

Options

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

Explanation

To solve the problem, we need to determine the truth values of \( p \), \( q \), and \( r \) given that the statement \( (p \land \neg r) \to (\neg p \lor q) \) is false.

### Step-by-Step Solution:

1. **Understanding the Implication**:
The implication \( A \to B \) is false only when \( A \) is true and \( B \) is false. Therefore, we need to find conditions where:
- \( A = (p \land \neg r) \) is true
- \( B = (\neg p \lor q) \) is false

2. **Finding Conditions for \( A \)**:
For \( A = (p \land \neg r) \) to be true:
- \( p \) must be true.
- \( \neg r \) must be true, which means \( r \) must be false.

So, we have:
- \( p = \text{True} \)
- \( r = \text{False} \)

3. **Finding Conditions for \( B \)**:
For \( B = (\neg p \lor q) \) to be false:
- Both \( \neg p \) and \( q \) must be false.
- Since \( p \) is true, \( \neg p \) is false.
- Therefore, \( q \) must also be false.

So, we have:
- \( q = \text{False} \)

4. **Conclusion**:
From the above analysis, we have determined the truth values:
- \( p = \text{True} \)
- \( q = \text{False} \)
- \( r = \text{False} \)

Thus, the truth values of \( p, q, \) and \( r \) are respectively: **True, False, False**.

### Summary of Truth Values:
- \( p = \text{True} \)
- \( q = \text{False} \)
- \( r = \text{False} \)