The problem asks us to analyze the relation $\rho$ defined on the set $\mathbb{Z}$ (the set of integers), where $\rho = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} \ | \ x \text{ divides } y\}$. We need to determine the properties of this relation—whether it is reflexive, symmetric, or both.

Analysis

1. Reflexivity: A relation is reflexive if for every $x \in \mathbb{Z}$, $(x, x) \in \rho$.

   • For any integer $x$, $x$ divides $x$ (since $x/x = 1$, which is an integer).
   • Hence, $\rho$ is reflexive.

2. Symmetry: A relation is symmetric if $(x, y) \in \rho$ implies $(y, x) \in \rho$.

   • If $x$ divides $y$, it does not necessarily follow that $y$ divides $x$. For example, $2 \mid 6$, but $6 \nmid 2$.
   • Hence, $\rho$ is not symmetric.

3. Conclusion: Since $\rho$ is reflexive but not symmetric, the correct answer is:

(A) Reflexive

Would you like further clarification or a breakdown of the definitions? Here are related questions to expand your understanding:

1. What is the definition of transitivity in relations, and is $\rho$ transitive?
2. How can we test if a relation is symmetric using specific examples?
3. What are some common relations on $\mathbb{Z}$ that are both reflexive and symmetric?
4. How does divisibility differ from equality in determining reflexivity or symmetry?
5. Could you extend the concept of divisibility to modular arithmetic relations?

Tip: To quickly check symmetry, use counterexamples involving non-equal integers.