STATEMENT-1 : `(phArrq)=~ p hArr q`
and
STATEMENT-2 : `(phArrq) =~p hArr ~q`

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 determine the validity of the statements given in the question, we will construct truth tables for both statements and compare their results.

### Step 1: Define the Statements
- **Statement 1**: \( (p \Leftrightarrow q) \equiv \neg p \Leftrightarrow q \)
- **Statement 2**: \( (p \Leftrightarrow q) \equiv \neg p \Leftrightarrow \neg q \)

### Step 2: Create the Truth Table
We will create a truth table for the variables \( p \) and \( q \) and evaluate both statements.

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

### Step 3: Evaluate Statement 1
- From the truth table, we see that:
- \( p \Leftrightarrow q \) is true when both \( p \) and \( q \) are true or both are false.
- \( \neg p \Leftrightarrow q \) is true when \( p \) is false and \( q \) is false, or when \( p \) is false and \( q \) is true.

Comparing the columns for \( p \Leftrightarrow q \) and \( \neg p \Leftrightarrow q \):
- They are not equal in all cases. Therefore, **Statement 1 is false**.

### Step 4: Evaluate Statement 2
- From the truth table, we see that:
- \( p \Leftrightarrow q \) is true when both \( p \) and \( q \) are true or both are false.
- \( \neg p \Leftrightarrow \neg q \) is true when both \( p \) and \( q \) are false or both are true.

Comparing the columns for \( p \Leftrightarrow q \) and \( \neg p \Leftrightarrow \neg q \):
- They are equal in all cases. Therefore, **Statement 2 is true**.

### Conclusion
- **Statement 1** is false.
- **Statement 2** is true.

### Final Answer
- The correct answer is that Statement 1 is false and Statement 2 is true.

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