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Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a is less than or equal to b to the power 2 } is neither reflexive nor symmetric nor transitive.

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.

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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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