If `p rarr (q vv r)` is false, then the truth values of p, q, r are respectively

Options

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

Answer

To solve the problem, we need to analyze the logical expression \( p \rightarrow (q \lor r) \) and determine the truth values of \( p \), \( q \), and \( r \) given that this expression is false.

### Step-by-Step Solution:

1. **Understanding the Implication**:
The expression \( p \rightarrow (q \lor r) \) is an implication. An implication \( A \rightarrow B \) is only false when \( A \) is true and \( B \) is false.

**Hint**: Recall the truth table for implications to understand when they are false.

2. **Setting Up the Condition**:
Since \( p \rightarrow (q \lor r) \) is false, we can conclude:
- \( p \) must be true (T).
- \( q \lor r \) must be false (F).

**Hint**: Remember that \( q \lor r \) (read as "q or r") is false only when both \( q \) and \( r \) are false.

3. **Analyzing \( q \lor r \)**:
For \( q \lor r \) to be false:
- \( q \) must be false (F).
- \( r \) must also be false (F).

**Hint**: Think about the logical OR operation; it is only false when both operands are false.

4. **Conclusion**:
From the above analysis, we can summarize the truth values:
- \( p \) is true (T).
- \( q \) is false (F).
- \( r \) is false (F).

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

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