Statement 1 : Consider the statements
p : Delhi is in India
q : Mumbai is not in Italy
Then the negation of the statement `p vv q`, is 'delhi is not in India and Mumbai is in Italy'
Statement 2: For any two statement p and `q ~(p vv q) = ~p vv ~q`

Options

A. Statement 1 and statement 2 are both false
B. Statement 1 and staement 2 are both true
C. statement 1 is true and statement 2 is false
D. statement 1 is false and statement 2 is true

Answer

To solve the problem, we need to analyze both statements provided and determine their truth values based on logical reasoning.

### Step-by-Step Solution:

1. **Identify the Statements**:
- Let \( p \): "Delhi is in India".
- Let \( q \): "Mumbai is not in Italy".

2. **Construct the Compound Statement**:
- The compound statement is \( p \lor q \) (read as "p or q"), which means "Delhi is in India or Mumbai is not in Italy".

3. **Negate the Compound Statement**:
- The negation of \( p \lor q \) is denoted as \( \sim (p \lor q) \).
- According to De Morgan's Laws, the negation of a disjunction is the conjunction of the negations:
\[ \sim (p \lor q) = \sim p \land \sim q \]

4. **Negate Each Statement**:
- Negating \( p \): \( \sim p \): "Delhi is not in India".
- Negating \( q \): \( \sim q \): "Mumbai is in Italy".

5. **Combine the Negated Statements**:
- Thus, \( \sim (p \lor q) = \sim p \land \sim q \) translates to:
- "Delhi is not in India and Mumbai is in Italy".

6. **Conclusion for Statement 1**:
- The negation of the statement \( p \lor q \) is indeed "Delhi is not in India and Mumbai is in Italy", which confirms that Statement 1 is **True**.

7. **Evaluate Statement 2**:
- Statement 2 claims: \( \sim (p \lor q) = \sim p \lor \sim q \).
- From De Morgan's Laws, we know that:
\[ \sim (p \lor q) = \sim p \land \sim q \]
- Therefore, Statement 2 is incorrect because it states that the negation of a disjunction is a disjunction of the negations, which is not true.

8. **Final Conclusion**:
- Statement 1 is **True** and Statement 2 is **False**.

### Final Answer:
- Statement 1 is true.
- Statement 2 is false.