show that the relation R on R defined as R={(a,b):a<_b } is reflexive, transitive but not symmetric

Chapter: Relations and functions 

Asked by niharikapabba2605 | 12th Oct, 2018, 12:43: PM

Expert Answer:

R = {(a, b); a ≤ b}

Obviously, (a, a) ∈ R as a = a.

Hence, R is reflexive.

Now,

(1, 2) ∈ R (as 1 < 2)

But, (2, 1) ∉ R as 2 is greater than 1.

Hence, R is not symmetric.

  Let (ab), (bc) ∈ R.

Then,

a ≤ b and b ≤ c

⇒ a ≤ c

⇒ (a, c) ∈ R

Hence, R is transitive.

Hence, R is reflexive and transitive but not symmetric.

 

Answered by Sneha shidid | 15th Oct, 2018, 09:51: AM