# Mark the correct alternative in the following question:

Question:

Mark the correct alternative in the following question:

The relation $S$ defined on the set $\mathbf{R}$ of all real number by the rule $a S b$ iff $a \geq b$ is

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

Solution:

We have,

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

As, $a=a \forall a \in \mathbf{R}$

$\Rightarrow(a, a) \in S$

So, $S$ is reflexive relation

Let $(a, b) \in S$

$\Rightarrow a \geq b$

But $b \leq a$

$\Rightarrow(b, a) \notin S$

So, $S$ is not symmetric relation

Let $(a, b) \in S$ and $(b, c) \in S$

$\Rightarrow a \geq b$ and $b \geq c$

$\Rightarrow a \geq \mathrm{c}$

$\Rightarrow(a, c) \in S$

So, $S$ is transitive relation

Hence, the correct alternative is option (b).