Question

$R$ is a relation over the set of real numbers and it is given by $mn\ge 0$ where $m,n$ are real numbers. Then $R$ is

• A. symmetric and transitive, not reflexive
• B. reflexive and symmetric, not transitive
• C. a partial order relation
• D. an equivalence relation

Answer 1

Answer: (D)

an equivalence relation

Checking for reflexive
For $m=n,$ $m^2$ is always $\ge 0$
So, $R$ is reflexive.
If $mn\ge 0$, then $nm\ge 0$
So, $R$ is symmetric.
If $mn\ge 0$ and $nk\ge 0$
$mn\ge 0$, if and only if $m$ and $n$ both have same sign

Also, $kn\ge 0$, if and only if $n$ and $k$ both have same sign

Thus, $m$ and $k$ both have same sign

$\therefore km\ge 0$ if $mn\ge 0$ and $nk\ge 0$
Therefore, $R$ is transitive.
Hence, $R$ is an equivalence relation.