Suppose p is a relation defined on the set Z,

p = {(x,y) EZxZ)(xy)is divisible by 3), then pis

(A) Reflexive but not Symmetric, Transitive

(B) Reflexive, Symmetric but not Transitive

(C) Reflexive. Transitive but not Symmetric (D) Equivalence

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.

Suppose p is a relation defined on the set Z, p = {(x,y) E Z x Z | (xy) is divisible by 3}, then p is: (A) Reflexive but not Symmetric, Transitive (B) Reflexive, Symmetric but not Transitive (C) Reflexive, Transitive but not Symmetric (D) Equivalence

Explore the relation p defined on the set of integers Z, where p = {(x, y) in Z x Z | (xy) is divisible by 3}. This problem delves into the properties of relations, including reflexivity, symmetry, and transitivity. Analyze whether p satisfies these properties and learn why the correct answer is Reflexive, Symmetric but not Transitive. Suitable for Grades 9-12.

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