Question

$S$ is a relation over the set $R$ of all real numbers and it is given by $(a,b)\in S\iff ab\ge 0$. Then, $S$ is

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

Answer 1

Answer: (D)

an equivalence relation

Reflexive relation:

$(a,a)\in S\iff a^2\ge 0$

Symmetric relation:

$(a,b)\in S\iff ab\ge 0$

$(b,a)\in S\iff ba\ge 0$

$\Rightarrow ab\ge 0$ when $a>0,b>0$ or $a<0,b<0$

Transitive relation:

$(a,b)\in S\iff ab\ge 0$

$(b,c)\in S\iff bc\ge 0$
$\Rightarrow a{b}^{2}c\ge 0$ [Taking product of above two equations]
$\Rightarrow ac\ge 0$

$\Rightarrow (a,c)\in S\iff ac\ge 0$

$\therefore S$ is an equivalence relaion.