Let p be the statement "It rains"! and q be the statement "It is cold".
STATEMENT-1 : If it rains then it is cold, if it does not rain then it is not cold.
and
STATEMENT-2 : `p rArr q =~ pvvq`

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 False, Statement-2 is True
D. Statement-1 is True, Statement-2 is False

Answer

To solve the problem, we need to analyze the two statements given in the question and verify their truth values using logical reasoning and truth tables.

### Step-by-Step Solution:

1. **Define the Statements**:
- Let \( p \) represent the statement "It rains".
- Let \( q \) represent the statement "It is cold".

2. **Translate Statement-1 into Logical Form**:
- Statement-1 says: "If it rains then it is cold, if it does not rain then it is not cold."
- This can be expressed in logical form as:
- \( p \rightarrow q \) (If it rains, then it is cold)
- \( \neg p \rightarrow \neg q \) (If it does not rain, then it is not cold)

3. **Construct the Truth Table for Statement-1**:
- We will create a truth table for \( p \) and \( q \) and evaluate \( p \rightarrow q \) and \( \neg p \rightarrow \neg q \).

| \( p \) | \( q \) | \( p \rightarrow q \) | \( \neg p \) | \( \neg q \) | \( \neg p \rightarrow \neg q \) |
|---------|---------|-----------------------|--------------|--------------|---------------------------------|
| T | T | T | F | F | T |
| T | F | F | F | T | T |
| F | T | T | T | F | F |
| F | F | T | T | T | T |

- From the truth table, we can see that:
- \( p \rightarrow q \) is true in 3 out of 4 cases.
- \( \neg p \rightarrow \neg q \) is true in 3 out of 4 cases.

4. **Evaluate Statement-1**:
- Statement-1 is true if both \( p \rightarrow q \) and \( \neg p \rightarrow \neg q \) are true.
- Since \( p \rightarrow q \) is false when \( p \) is true and \( q \) is false, and \( \neg p \rightarrow \neg q \) is false when \( p \) is false and \( q \) is true, we conclude that Statement-1 is not universally true.

5. **Translate Statement-2 into Logical Form**:
- Statement-2 states: \( p \rightarrow q \equiv \neg p \vee q \).
- This means "If it rains, then it is cold" is logically equivalent to "Either it does not rain or it is cold".

6. **Construct the Truth Table for Statement-2**:
- We will create a truth table for \( \neg p \vee q \).

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \vee q \) |
|---------|---------|--------------|----------------------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

7. **Evaluate Statement-2**:
- From the truth table, we see that \( p \rightarrow q \) and \( \neg p \vee q \) yield the same truth values:
- Both are true in 3 out of 4 cases and false in 1 case.
- Therefore, Statement-2 is true.

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

### Final Answer:
- **Statement-1**: False
- **Statement-2**: True