Question

Show that the relation R in R defined as R = {( a , b ): a ≤ b }, is reflexive and transitive but not symmetric.

Answer 1

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

( a,a )∈R, since, for all values of a, a∈R, a≤a. Hence, R is reflexive.

Let, ( a,b )∈R, therefore a≤b. Then ( b,a )∉R since b≥a. Hence, Ris not symmetric.

Let, ( a,b ),( b,c )∈R, thus a≤b and b≤c. So, a≤c implies that ( a,c )∈R. Hence, Ris transitive.

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