Let N be the set of integers. A relation R on N is defined as `R = {(x, y) | xy gt 0, xy in N}`.
Then, which one of the following is correct?

Options

A. R is symmetric but not reflexive
B. R is reflexive but not symmetric
C. R is symmetric and reflexive but not transitive
D. R is an equivalence relation

Answer

To solve the problem, we need to analyze the relation \( R \) defined on the set of integers \( N \) as follows:

\[ R = \{(x, y) \mid xy > 0, \, x, y \in N\} \]

### Step 1: Understanding the Relation
The relation \( R \) consists of pairs of integers \( (x, y) \) such that the product \( xy \) is greater than 0. This means both \( x \) and \( y \) must either be positive or both must be negative.

### Step 2: Reflexivity
A relation is reflexive if every element is related to itself. For \( R \) to be reflexive, we need to check if \( (x, x) \in R \) for all \( x \in N \).

- For any integer \( x \), \( x \cdot x = x^2 \) which is always greater than 0 if \( x \) is a positive integer. Thus, \( (x, x) \in R \).

**Conclusion:** \( R \) is reflexive.

### Step 3: Symmetry
A relation is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \).

- If \( xy > 0 \), then both \( x \) and \( y \) must be either positive or negative. Therefore, \( yx = xy > 0 \) implies \( (y, x) \in R \).

**Conclusion:** \( R \) is symmetric.

### Step 4: Transitivity
A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).

- Suppose \( (x, y) \in R \) and \( (y, z) \in R \). This means \( xy > 0 \) and \( yz > 0 \).
- If \( xy > 0 \), both \( x \) and \( y \) are either positive or negative. Similarly, if \( yz > 0 \), both \( y \) and \( z \) are either positive or negative.
- Hence, if \( y \) is positive, then both \( x \) and \( z \) must also be positive; if \( y \) is negative, then both \( x \) and \( z \) must also be negative. Thus, \( xz > 0 \).

**Conclusion:** \( R \) is transitive.

### Final Conclusion
Since the relation \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Answer
The correct option is **D**: The relation \( R \) is an equivalence relation.

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