Question

Mark the correct alternative in the following question:

The relation S defined on the set R of all real number by the rule aSb iff a $\ge$ b is

(a) an equivalence relation
(b) reflexive, transitive but not symmetric
(c) symmetric, transitive but not reflexive
(d) neither transitive nor reflexive but symmetric

Answer 1

We have,

S = {(a, b) : a $\ge$ b; a, b $\in$ R}

$$\begin{gathered} \mathrm{As},a=a\forall a\in \mathbf{R} \\ \Rightarrow \left(a,a\right)\in S \\ \mathrm{So},S\mathrm{is}\mathrm{reflexive}\mathrm{relation} \\ \mathrm{Let}\left(a,b\right)\in S \\ \Rightarrow a\ge b \\ \mathrm{But}b\le a \\ \Rightarrow \left(b,a\right)\notin S \\ \mathrm{So},S\mathrm{is}\mathrm{not}\mathrm{symmetric}\mathrm{relation} \\ \mathrm{Let}\left(a,b\right)\in S\mathrm{and}\left(b,c\right)\in S \\ \Rightarrow a\ge b\mathrm{and}b\ge c \\ \Rightarrow a\ge \mathrm{c} \\ \Rightarrow \left(a,c\right)\in S \\ \mathrm{So},S\mathrm{is}\mathrm{transitive}\mathrm{relation} \end{gathered}$$

Hence, the correct alternative is option (b).