Q.2 Show that the relation R in the set R of real numbers, defined as $R = \{(a, b) : a \leq b^2 \}$ is neither reflexive nor symmetric nor transitive.

$R = \{(a, b) : a \leq b^2 \}$

Taking

$\left ( \frac{1}{2},\frac{1}{2} \right )\notin R$

and

$\left ( \frac{1}{2} \right )> \left ( \frac{1}{2} \right )^{2}$

So,R is not reflexive.

Now,

$\left ( 1,2 \right )\in R$ because    $1< 4$.

But, $4\nless 1$  i.e. 4 is not less than 1

So,$\left ( 2,1 \right )\notin R$

Hence, it is not symmetric.

$\left ( 3,2 \right )\in R\, \, and \, \, \left ( 2,1.5 \right )\in R$   as $3< 4\, \, and \, \, 2< 2.25$

Since $\left ( 3,1.5 \right )\notin R$  because $3\nless 2.25$

Hence, it is not transitive.

Thus, we can conclude that it is neither reflexive, nor symmetric ,nor transitive.

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