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 if y is a transcendental number.
Statement-1 : r is equivalent to either p or q.
Statement-2: r is equivalent to `~{p hArr ~q)`.

Options

A. Statement-1 is true, statement-2 is true, statement-2 is correct explanation for statement-0
B. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-0
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 statements given and determine their logical equivalences.

### Step-by-Step Solution:

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

2. **Expressing Statement \( r \)**:
- The statement \( r \) can be expressed in logical terms as:
\[ r: q \implies \neg p \]
- This means if \( y \) is a transcendental number (i.e., \( q \) is true), then \( x \) must be a rational number (i.e., \( p \) is false).

3. **Understanding the Logical Equivalence**:
- The statement \( r \) can also be rewritten using the contrapositive:
\[ r: \neg p \implies \neg q \]
- This means if \( x \) is not a rational number (i.e., \( x \) is irrational), then \( y \) is not a transcendental number.

4. **Analyzing Statement 1**:
- Statement 1 claims that \( r \) is equivalent to either \( p \) or \( q \):
\[ r \equiv p \lor q \]
- This is incorrect because \( r \) does not imply that either \( p \) or \( q \) must be true; it states a conditional relationship.

5. **Analyzing Statement 2**:
- Statement 2 claims that \( r \) is equivalent to \( \neg p \iff \neg q \):
\[ r \equiv \neg p \iff \neg q \]
- This is correct because it captures the logical relationship established in step 3.

6. **Conclusion**:
- Therefore, Statement 1 is false, and Statement 2 is true.

### Final Answer:
- Statement 1 is false, and Statement 2 is true.