Given the relation $p = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid xy \text{ is divisible by 3}\}$ defined on the set of integers $\mathbb{Z}$, we need to determine the properties of this relation—whether it is reflexive, symmetric, transitive, or an equivalence relation.

1. Reflexive:

A relation $p$ is reflexive if $(x, x) \in p$ for all $x \in \mathbb{Z}$, meaning $x \cdot x$ is divisible by 3.

• For $p$ to be reflexive, we need $x \cdot x$ to be divisible by 3 for every $x \in \mathbb{Z}$.
• However, not every integer satisfies $x^2$ divisible by 3. Only integers divisible by 3 satisfy this condition.

Thus, p is not reflexive.

2. Symmetric:

A relation $p$ is symmetric if whenever $(x, y) \in p$, then $(y, x) \in p$ as well, i.e., if $xy$ is divisible by 3, then $yx$ must also be divisible by 3.

• Since multiplication is commutative ($xy = yx$), if $xy$ is divisible by 3, so is $yx$.

Thus, p is symmetric.

3. Transitive:

A relation $p$ is transitive if whenever $(x, y) \in p$ and $(y, z) \in p$, then $(x, z) \in p$ as well, i.e., if $xy$ and $yz$ are divisible by 3, then $xz$ should also be divisible by 3.

• This does not always hold. For instance, if $x = 3$, $y = 1$, and $z = 3$, then $x \cdot y = 3$ (divisible by 3) and $y \cdot z = 3$ (divisible by 3), but $x \cdot z = 9$ is divisible by 3. However, if $y = 2$, neither $x \cdot z$ nor $x \cdot y$ might be divisible by 3, breaking the transitivity.

Thus, p is not transitive.

Conclusion:

From the analysis, the relation is symmetric but neither reflexive nor transitive. Therefore, the correct answer is:

(B) Reflexive, Symmetric but not Transitive.

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Would you like any additional clarification or details on this problem? Here are a few related questions for further practice:

1. What is an example of a reflexive relation that is also symmetric?
2. Can you provide a set where the relation $p$ would be reflexive?
3. What is the difference between transitive and symmetric relations?
4. If a relation is reflexive, symmetric, and transitive, what is it called?
5. How does the property of symmetry relate to commutative operations?

Tip: Always check symmetry and reflexivity using small examples for clarity, especially when dealing with multiplication or divisibility relations.