Question

S is a relation over the set R of all real numbers and it is given by
(a, b) ∈ S ⇔ ab ≥ 0. Then, S is
(a) symmetric and transitive only
(b) reflexive and symmetric only
(c) antisymmetric relation
(d) an equivalence relation

Answer 1

(d) an equivalence relation

Reflexivity: Let a$\in$R
Then,

$$\begin{gathered} aa=a^2>0 \\ \Rightarrow \left(a,a\right)\in R\forall a\in R \end{gathered}$$

So, S is reflexive on R.

Symmetry: Let (a, b)$\in$S
Then,

$$\begin{gathered} \left(a,b\right)\in S \\ \Rightarrow ab\ge 0 \\ \Rightarrow ba\ge 0 \\ \Rightarrow \left(b,a\right)\in S\forall a,b\in R \end{gathered}$$

So, S is symmetric on R.

Transitivity:

$$\begin{gathered} \mathrm{If}\left(a,b\right),\left(b,c\right)\in \mathrm{S} \\ \Rightarrow ab\ge 0\mathrm{and}bc\ge 0 \\ \Rightarrow ab\times bc\ge 0 \\ \Rightarrow ac\ge 0\left[\because {\mathrm{b}}^2\ge 0\right] \\ \Rightarrow \left(a,c\right)\in S\mathrm{for}\mathrm{all}a,b,c\in \mathrm{set}R \end{gathered}$$

Hence, S is an equivalence relation on R.