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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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