Consider the following two propositions :
`P1:~(p rarr ~ q)`
`P2: (p^^~q) ^^ ((-p) vvq)`
If the proposition `p rarr ((~p) vv q)` is evaluated as FALSE, then ,

Options

A. P1 is TRUE and P2 is FALSE
B. P1 is FALSE and P2 is TRUE
C. Both P1 and P2 are FALSE
D. Both P1 and P2 are TRUE

Answer

To solve the problem, we need to analyze the propositions given and determine the truth values based on the condition that \( p \rightarrow (\neg p \lor q) \) is evaluated as FALSE.

### Step-by-Step Solution:

1. **Understanding the Implication**:
The implication \( p \rightarrow (\neg p \lor q) \) is FALSE only when the antecedent (the part before the arrow) is TRUE and the consequent (the part after the arrow) is FALSE. Therefore, we have:
- \( p \) is TRUE
- \( \neg p \lor q \) is FALSE

2. **Analyzing \( \neg p \lor q \)**:
The expression \( \neg p \lor q \) is FALSE when both \( \neg p \) and \( q \) are FALSE. Since \( p \) is TRUE:
- \( \neg p \) is FALSE
- \( q \) must also be FALSE

3. **Values of \( p \) and \( q \)**:
From the above analysis, we conclude:
- \( p = \text{TRUE} \)
- \( q = \text{FALSE} \)

4. **Evaluating Propositions \( P1 \) and \( P2 \)**:
- **For \( P1: \neg(p \rightarrow \neg q) \)**:
- First, evaluate \( p \rightarrow \neg q \):
- Since \( p \) is TRUE and \( q \) is FALSE, \( \neg q \) is TRUE.
- Therefore, \( p \rightarrow \neg q \) is TRUE (because TRUE implies TRUE).
- Now, \( \neg(p \rightarrow \neg q) \) is FALSE (since the negation of TRUE is FALSE).

- **For \( P2: (p \land \neg q) \land (\neg p \lor q) \)**:
- Evaluate \( p \land \neg q \):
- \( p \) is TRUE and \( \neg q \) is TRUE, so \( p \land \neg q \) is TRUE.
- Evaluate \( \neg p \lor q \):
- \( \neg p \) is FALSE and \( q \) is FALSE, so \( \neg p \lor q \) is FALSE.
- Now, \( (p \land \neg q) \land (\neg p \lor q) \) is TRUE AND FALSE, which is FALSE.

5. **Final Values**:
- \( P1 \) is FALSE.
- \( P2 \) is FALSE.

### Conclusion:
Both \( P1 \) and \( P2 \) are FALSE.

### Answer:
Both \( P1 \) and \( P2 \) are false.

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