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# Check whether the relation R in R defined by R = {(a, b) : a is less than or equal to b to the power 3 } is reflexive, symmetric or transitive.

Q.5 Check whether the relation R in R defined by $R = \{(a, b) : a \leq b^3 \}$ is reflexive,
symmetric or transitive.

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$R = \{(a, b) : a \leq b^3 \}$

$\left ( \frac{1}{2},\frac{1}{2} \right )\notin R$     because     $\frac{1}{2}\nleqslant (\frac{1}{2}) ^{3}$

So, it is not symmetric

Now, $\left ( 1,2 \right ) \in R$    because $1< 2^{3}$

but $\left ( 2,1 \right )\notin R$  because $2\nleqslant 1^{3}$

It is not symmetric

$\left ( 3,1.5 \right ) \in R\, \, and \, \, \left ( 1.5,1.2 \right ) \in R$   as   $3< 1.5^{3} \, \, and \, \, 1.5< 1.2^{3}$.

But, $\left ( 3,1.2 \right )\notin R$   because $3 \nleqslant 1.2^{3}$

So it is not transitive

Thus, it is neither reflexive, nor symmetric,nortransitive.

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