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
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).
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