Let N be the set of integers. A relation R or N is defined as `R={(x,y):xygt0, x,y,inN}`. 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

Explanation

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

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

### Step 1: Understand the Relation
The relation \( R \) consists of pairs \( (x, y) \) where the product \( xy \) is greater than 0 and both \( x \) and \( y \) are natural numbers. Since natural numbers are always positive, the product \( xy \) will always be greater than 0 for any \( x, y \in N \).

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

\[ (x, x) \in R \text{ for all } x \in N \]

Since \( x \in N \) implies \( x > 0 \), we have \( xx = x^2 > 0 \). Therefore, \( (x, x) \in R \) for all \( x \in N \). Thus, \( R \) is reflexive.

### Step 3: Check for Symmetry
A relation is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \). For \( R \), if \( (x, y) \in R \), it means \( xy > 0 \). Since multiplication is commutative, \( yx = xy > 0 \) also holds. Hence, \( (y, x) \in R \). Therefore, \( R \) is symmetric.

### Step 4: Check for Transitivity
A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). For \( R \), assume \( (x, y) \in R \) and \( (y, z) \in R \). This means:

1. \( xy > 0 \)
2. \( yz > 0 \)

Since \( x, y, z \in N \) (and thus positive), both conditions imply that \( x > 0 \), \( y > 0 \), and \( z > 0 \). Therefore, \( xz = x \cdot z > 0 \) holds true, which means \( (x, z) \in R \). Hence, \( R \) is transitive.

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

### Final Answer
The correct option is that \( R \) is an equivalence relation.

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