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

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Solution

The given relation R in set R of real numbers is defined as R={( a,b ):a≤ b 2 }.

( 1 2 , 1 2 )∉R, since, ( 1 2 )> ( 1 2 ) 2 =( 1 4 ), hence R is not reflexive.

( 1,3 )∈R, since, 1< 3 2 =9. But ( 3,1 )∉R since 3> 1 2 =1. Hence, Ris not symmetric.

( 3,2 )and ( 2,1.5 )∈R. Since 3< 2 2 =4 and 2< 1.5 2 =2.25. But ( 3,1.5 )∉R, since 3> 1.5 2 =2.25. Hence, Ris not transitive.

Therefore, the given relation R={( a,b ):a≤ b 2 } in the set R of real numbers is not reflexive, symmetric, or transitive.

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