STATEMENT-1 : `(p^^~q)^^(~p^^q)`
and
STATEMENT-2 : `(pvv~q) vv (~p ^^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 analyze the given statements, we will evaluate the truth values of both statements step by step.

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

### Step 2: Create a Truth Table
We will create a truth table to evaluate the truth values of both statements based on the possible values of \(p\) and \(q\).

| \(p\) | \(q\) | \(\neg p\) | \(\neg q\) | \(p \land \neg q\) | \(\neg p \land q\) | Statement 1 | \(p \lor \neg q\) | Statement 2 |
|-------|-------|------------|------------|---------------------|---------------------|--------------|--------------------|--------------|
| T | T | F | F | F | F | F | T | T |
| T | F | F | T | T | F | F | T | T |
| F | T | T | F | F | T | F | T | T |
| F | F | T | T | F | F | F | T | T |

### Step 3: Evaluate Statement 1
- For Statement 1: \((p \land \neg q) \land (\neg p \land q)\)
- The conjunction is true only if both parts are true.
- From the truth table, we see that the value of Statement 1 is **False** for all combinations of \(p\) and \(q\).

### Step 4: Evaluate Statement 2
- For Statement 2: \((p \lor \neg q) \lor (\neg p \land q)\)
- The disjunction is true if at least one part is true.
- From the truth table, we see that the value of Statement 2 is **True** for all combinations of \(p\) and \(q\).

### Conclusion
- **Statement 1** is **False**.
- **Statement 2** is **True**.

### Final Result
Thus, we conclude that Statement 1 is not equivalent to Statement 2.

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