Question

$R$ is a relation over the set of real numbers and it is given by $nm\ge 0$. Then $r$ is

• A. Symmetric and transitive
• B. Reflexive and symmetric
• C. A partial order relation
• D. An equivalence relation

Answer 1

The correct option is D

An equivalence relation

Explanation for the correct answer:

Checking if the relation is reflexive, transitive, and symmetric:

$mRniffmn\ge 0$

Checking for reflexive property:

It is known that $m^2\ge 0$

$mm\ge 0$

$mRm$

Therefore, It is reflexive.

Checking for symmetric property:

Let $mRn$

$$\begin{gathered} mn\ge 0 \\ nm\ge 0 \\ nRm \end{gathered}$$

Therefore, It is symmetric.

Checking for transitive property:

Let $mRn,nRp$

$mn\ge 0;np\ge 0$

$m,n,p$ are of the same signs

$mp\ge 0$

$mRp$

Therefore, It is transitive.

Hence, It is an equivalence relation.

Hence , the correct answer is option (D).