If $$ x $$ is a rational number and $$ y $$ is an irrational number, then which of the following statements is true?

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To analyze the given statements:

1. **Statement (a):**  
   If $$ x = 0 $$ (rational) and $$ y = \sqrt{2} $$ (irrational), then:  
   $$ x + y = 0 + \sqrt{2} = \sqrt{2} $$, which is irrational.  
   Hence, $$ x + y $$ is not always rational. This statement is **false**.

2. **Statement (b):**  
   If $$ x = 0 $$ (rational) and $$ y = \sqrt{2} $$ (irrational), then:  
   $$ xy = 0 	imes \sqrt{2} = 0 $$, which is rational.  
   Thus, $$ xy $$ is not necessarily irrational. This statement is **false**.

3. **Statement (c):**  
   As shown in (b), $$ xy $$ can be rational (e.g., when $$ x = 0 $$) or irrational (e.g., when $$ x = 1, y = \sqrt{2} $$). Therefore, it is not always irrational. This statement is **true**.

4. **Statement (d):**  
   - The sum of a rational number ($$ x $$) and an irrational number ($$ y $$) is always irrational because adding a non-zero rational number cannot "cancel out" the irrationality of $$ y $$.  
   - The product of a rational number ($$ x $$) and an irrational number ($$ y $$) can be either rational (e.g., if $$ x = 0 $$) or irrational (e.g., if $$ x = 1, y = \sqrt{2} $$).

Thus, this statement is **true**.

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