Question

Check whether the relation R in R defined as R = {( a , b ): a ≤ b 3 } is reflexive, symmetric or transitive.

Answer 1

The given relation R in set R of real numbers is defined as R={( a,b ):a≤ b 3 }.

( 1 2 , 1 2 )∉R, since, 1 2 > ( 1 2 ) 3 = 1 8 . Hence, R is not reflexive.

( 1,2 )∈R, since 1< 2 3 =8, ( 2,1 )∉R and 2> 1 3 =1. Hence, Ris not symmetric.

( 3, 3 2 ),( 3 2 , 6 5 )∈R, since 3< ( 3 2 ) 3 = 9 8 =1.125 and 3 2 < ( 6 5 ) 3 = 216 125 =1.728 but ( 3, 6 5 )∉R, since 3> ( 6 5 ) 3 = 216 125 =1.728 , hence Ris not transitive.

Therefore, the given relation R={( a,b ):a≤ b 3 } in set R of real numbers is not reflexive, symmetric, or transitive.