Let p be the statement "x is an irrational number", q be the statement "y is a transcendental number " and r be the statement "x is a rational number iff y is a transcendental number".
Statement - 1 : r is equivalent to either q or p.
Statement -2 : r is equivalent to `~(p harr ~q)`

Options

A. S -1 is true , S - 2 is true, S -2 is a correct explanation of S-1
B. S-1 is true, S - 2 is true, S - 2 is not a correct explanation of S-1
C. S-1 is true, S - 2 is false
D. S-1 is false, S - 2 is true

Answer

To solve the problem, we need to analyze the statements given and determine their truth values based on the definitions of the statements \( p \), \( q \), and \( r \).

### Step-by-step Solution:

1. **Define the Statements**:
- Let \( p \): "x is an irrational number"
- Let \( q \): "y is a transcendental number"
- Let \( r \): "x is a rational number iff y is a transcendental number"

The statement \( r \) can be rewritten in logical terms as:
\[ r \equiv \neg p \iff q \]
This means that \( r \) is true if both \( \neg p \) (x is a rational number) and \( q \) (y is a transcendental number) have the same truth value.

2. **Truth Table Setup**:
We will create a truth table for \( p \), \( q \), \( \neg p \), and \( r \).

| \( p \) | \( q \) | \( \neg p \) | \( r \) ( \( \neg p \iff q \) ) |
|---------|---------|---------------|---------------------------------|
| T | T | F | F |
| T | F | F | T |
| F | T | T | T |
| F | F | T | F |

- **Explanation of \( r \)**:
- When \( p \) is true and \( q \) is true, \( r \) is false (because \( \neg p \) is false).
- When \( p \) is true and \( q \) is false, \( r \) is true (both \( \neg p \) and \( q \) are false).
- When \( p \) is false and \( q \) is true, \( r \) is true (both \( \neg p \) and \( q \) are true).
- When \( p \) is false and \( q \) is false, \( r \) is false (because \( \neg p \) is true and \( q \) is false).

3. **Evaluate Statement 1**: \( r \equiv q \lor p \)
- We will check the truth values of \( r \) and \( q \lor p \).

| \( p \) | \( q \) | \( q \lor p \) | \( r \) | \( r \equiv (q \lor p) \) |
|---------|---------|----------------|---------|---------------------------|
| T | T | T | F | F |
| T | F | T | T | T |
| F | T | T | T | T |
| F | F | F | F | T |

- The equivalence \( r \equiv (q \lor p) \) is not satisfied in all cases. Therefore, **Statement 1 is false**.

4. **Evaluate Statement 2**: \( r \equiv \neg p \iff \neg q \)
- We will check the truth values of \( r \) and \( \neg p \iff \neg q \).

| \( p \) | \( q \) | \( \neg q \) | \( \neg p \) | \( \neg p \iff \neg q \) | \( r \) | \( r \equiv (\neg p \iff \neg q) \) |
|---------|---------|---------------|---------------|---------------------------|---------|-------------------------------------|
| T | T | F | F | T | F | F |
| T | F | T | F | F | T | F |
| F | T | F | T | F | T | F |
| F | F | T | T | T | F | F |

- The equivalence \( r \equiv (\neg p \iff \neg q) \) is also not satisfied in all cases. Therefore, **Statement 2 is false**.

5. **Conclusion**:
Both statements are false. Thus, the answer is that neither statement is correct.