Question

A relation R in set A = {1, 2, 3} is defined as:
R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (1, 3), (3, 1), (2, 3), (3, 2)}
Test the reflexivity, symmetricity and transitivity of R.

Answer 1

Given : Set A = {1, 2, 3}

Relation R in A is defined as

R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (1, 3), (3, 1) (2, 3), (3, 2)}

(i) Reflexivity : 

Here

(1, 1) ∈ R

(2, 2) ∈ R

(3, 3) ∈ R

So, ∀ a ∈ A ⇒ (a, a) ∈ R

R is not reflexive.

(ii) Symmetricity :

Here

(1, 2) ∈ R ⇔ (2, 1) ∈ R

(2, 3) ∈ R ⇔ (3, 2) ∈ R

(1, 3) ∈ R ⇔ (3, 1) ∈ R

So, (a, b) ∈ R ⇒ (b, a) ∈ R
R is symmetric relation.

(iii) Transitivity:

(1, 2) ∈ R, (2, 1) ∈ R ⇔ (1, 1) ∈ R etc.,

So, by definition of transitive relation.

R is transitive if

(a, b) ∈ R, (b, c) ∈ R

⇒ (a, c) ∈ R ∀ a, b, c ∈ A