S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab $\geq$ 0. Then, S is _______________ .

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KollegeApply · Accepted Answer

an equivalence relation Reflexivity: Let a ∈ R Then, aa = a2 > 0 $\Rightarrow$ (a, a) ∈ R ∀ a ∈ R So, S is reflexive on R. Symmetry: Let (a, b) ∈ S Then, (a, b) ∈ S $\Rightarrow$ ab $\geq$ 0 $\Rightarrow$ ba $\geq$ 0 $\Rightarrow$ (b, a) ∈ S ∀ a, b ∈ R So, S is symmetric on R. Transitive: If (a, b), (b, c) ∈ S $\Rightarrow$ ab $\geq$ 0 and bc $\geq$ 0 $\Rightarrow$ ab x bc $\geq$ 0 $\Rightarrow$ ac $\geq$ 0 [∵ b2 $\geq$ 0] $\Rightarrow$ (a, c) ∈ S for all a, b, c ∈ set R Hence, S is an equivalence relation on R

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