Question

If the relation is defined on $R$ by $(x,y)\in S\iff xy\ge 0$, then $S$ is

• A. an equivalence relation
• B. symmetric only
• C. reflexive only
• D. transitive only

Answer 1

The correct option is A

an equivalence relation

Explanation for the correct option.

Step 1: For reflexive

Given, a relation is defined on $R$ by $(x,y)\in S\iff xy\ge 0$.

For reflexive.

Let $\left(x,x\right)\in S$

$\Rightarrow x^2\ge 0$

So, the relation is reflexive.

Step 2: For symmetric

For symmetric.

Let $\left(x,y\right)\in S$ then $xy\ge 0$

$\Rightarrow yx\ge 0$

So, the relation is symmetric.

Step 3: For transitive

For transitive.

Let $\left(x,y\right)\in S$ then $xy\ge 0$ and $\left(y,z\right)\in S$ then $yz\ge 0$.

$\Rightarrow xz\ge 0$ i.e; $\left(x,z\right)\in S$

So, the relation is transitive.

As the relation is reflexive, symmetric & transitive so it is an equivalence relation.

Hence, option A is correct.