Question

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

• A. symmetric and transitive
• B. reflexive and symmetric
• C. a partial order relation
• D. an equivalence relation

Answer 1

Answer: (D)

an equivalence relation

Given , $mRn$ iff $mn\ge 0$
Reflexivity:
We know that $m^2\ge 0$
$\Rightarrow mm\ge 0$
$\Rightarrow mRm$
Hence, r is reflexive.
Symmetry:
Let $mRn$
$\Rightarrow mn\ge 0$
$\Rightarrow nm\ge 0$ (product of real numbers is commutative.)
$\Rightarrow nRm$
Hence, R is symmetric.
Transitive
Let $mRn,nRp$
$\Rightarrow mn\ge 0;np\ge 0$
$\Rightarrow m,n,p$ are of same signs
$\Rightarrow mp\ge 0$
$\Rightarrow mRp$
Hence, R is transitive.
Hence, R is an equivalence relation.