Question

Give an example of a relation:
(1) Reflexive but not symmetric
(2) Symmetric but not transitive
(3) Transitive but not symmetric
plz reply...it's urgent

Answer 1

1) Define a relation R in R as:

R = {a, b): a3 ≥ b3}

Clearly (a, a) ∈ R as a3 = a3.

∴ R is reflexive.

Now,

(2, 1) ∈ R (as 23 ≥ 13)

But,

(1, 2) ∉ R (as 13 < 23)

∴ R is not symmetric.



Hence, relation R is reflexive but not symmetric.

2)
Let A = {5, 6, 7}.

Define a relation R on A as R = {(5, 6), (6, 5)}.


Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric.

=> (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R

∴R is not transitive.

Hence, relation R is symmetric but not transitive.

3)

Consider a relation R in R defined as:

R = {(a, b): a < b}



Now,

(1, 2) ∈ R (as 1 < 2)

But, 2 is not less than 1.

∴ (2, 1) ∉ R

∴ R is not symmetric.

Now, let (a, b), (b, c) ∈ R.

⇒ a < b and b < c

⇒ a < c

⇒ (a, c) ∈ R

∴ R is transitive.

Hence, relation R is transitive but not symmetric.