STATEMENT-1 : `(p rArr q) hArr (~q rArr~p)`
and
STATEMENT-2 : `p rArr q` is logically equivalent to `~q rArr ~p`

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 problem, we need to analyze the two statements given and determine their logical equivalence. We will do this by constructing truth tables for both statements.

### Step 1: Define the Statements
- **Statement 1**: \((p \Rightarrow q) \Leftrightarrow (\neg q \Rightarrow \neg p)\)
- **Statement 2**: \(p \Rightarrow q\) is logically equivalent to \(\neg q \Rightarrow \neg p\)

### Step 2: Construct the Truth Table
We have two variables \(p\) and \(q\). The possible truth values for \(p\) and \(q\) are True (T) and False (F). Therefore, we can create a truth table with all combinations of \(p\) and \(q\).

| \(p\) | \(q\) | \(\neg p\) | \(\neg q\) | \(p \Rightarrow q\) | \(\neg q \Rightarrow \neg p\) | \((p \Rightarrow q) \Leftrightarrow (\neg q \Rightarrow \neg p)\) |
|-------|-------|------------|------------|----------------------|-------------------------------|-----------------------------------------------------|
| T | T | F | F | T | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |

### Step 3: Analyze the Truth Table
1. **Column for \(p \Rightarrow q\)**:
- True when \(p\) is True and \(q\) is True.
- False when \(p\) is True and \(q\) is False.
- True when \(p\) is False (regardless of \(q\)).

2. **Column for \(\neg q \Rightarrow \neg p\)**:
- True when \(\neg q\) is False (i.e., \(q\) is True).
- False when \(\neg q\) is True and \(\neg p\) is False (i.e., \(q\) is False and \(p\) is True).
- True when both \(\neg q\) and \(\neg p\) are True (i.e., both \(p\) and \(q\) are False).

3. **Column for \((p \Rightarrow q) \Leftrightarrow (\neg q \Rightarrow \neg p)\)**:
- This column is True when both previous columns have the same truth value.

### Step 4: Conclusion
From the truth table, we see that the biconditional \((p \Rightarrow q) \Leftrightarrow (\neg q \Rightarrow \neg p)\) is always True. Therefore, Statement 1 is a tautology.

### Step 5: Evaluate Statement 2
Statement 2 states that \(p \Rightarrow q\) is logically equivalent to \(\neg q \Rightarrow \neg p\). Since we have shown that both statements yield the same truth values, Statement 2 is also True.

### Final Answer
- **Statement 1** is True (it is a tautology).
- **Statement 2** is True (they are logically equivalent).