STATEMENT-1 : If `(p^^~r)rArr(qvvr)` is false and q and r are false , then p is also false.
and
STATEMENT-2 : `p rArr q` is false, when q is false and p is true. ·

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the given problem, we need to analyze both statements step by step.

**Statement 1: If (p ∧ ¬r) → (q ∨ r) is false and q and r are false, then p is also false.**

1. **Understanding the implications of the statement**:
- The expression (p ∧ ¬r) → (q ∨ r) is an implication. An implication is false only when the antecedent (the part before the arrow) is true and the consequent (the part after the arrow) is false.
- We know that q and r are both false.

2. **Substituting values**:
- Since q is false and r is false, we can evaluate the consequent:
- q ∨ r = false ∨ false = false.
- Now, we need to evaluate the antecedent (p ∧ ¬r):
- Since r is false, ¬r = true.
- Therefore, (p ∧ ¬r) = p ∧ true = p.

3. **Analyzing the implication**:
- For the implication (p ∧ ¬r) → (q ∨ r) to be false, we have:
- p must be true (because the antecedent must be true).
- The consequent (q ∨ r) is false.
- Thus, we have p = true and (q ∨ r) = false.

4. **Conclusion**:
- If p is true, then it contradicts the statement that p is also false. Therefore, Statement 1 is false.

**Statement 2: p → q is false when q is false and p is true.**

1. **Understanding the implication**:
- The implication p → q is false only when p is true and q is false.

2. **Substituting values**:
- We are given that p is true and q is false.

3. **Analyzing the implication**:
- If p is true and q is false, then:
- p → q = true → false = false.

4. **Conclusion**:
- Thus, Statement 2 is true.

**Final Conclusion**:
- Statement 1 is false.
- Statement 2 is true.

The correct answer is that Statement 1 is false and Statement 2 is true.