Question

The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on a set A = {1, 2, 3} is
1. reflexive transitive but not symmetric
2. reflexive, symmetric but not transitive
3. symmetric, transitive but not reflexive
4. reflexive but neither symmetric not transitive

Answer 1

Correct Answer - Option 1 : reflexive transitive but not symmetric

Concept:

Let A be a set in which the relation R defined. 

1.R is said to be a Reflexive Relation  (a, a) ∈ R

2. R is said to be a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R

3. R is said to be a transitive relation, if (a, b) ∈ R , (b, c) ∈ R ⇒ (a, c) ∈ R

 

 

Calculations:

Given set is A = {1, 2, 3}

and the relation is R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} 

Let A be a set in which the relation R defined.

1.R is said to be a Reflexive Relation  (a, a) ∈ R

2. R is said to be a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R

3. R is said to be a transitive relation, if (a, b) ∈ R , (b, c) ∈ R ⇒ (a, c) ∈ R

 

Since, 1, 2 , 3 ∈ A and (1, 1), (2, 2), (3, 3) \(\rm ∈ R\)

⇒ Every element maps to itself.

⇒ R is Reflexive 

Now, 1, 2 , 3 \(\rm ∈ R\)

(1, 2), (2, 3) \(\rm ∈ R\) ⇒  (1, 3) \(\rm ∈ R\)

⇒R relates 1 to 2 and 2 to 3, then R also relates 1 to 3

⇒ R is Transitive 

Here, R is not symmetric relation, as (a, b) ∈ R \(\neq \) (b, a) ∈ R

Hence, The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on a set A = {1, 2, 3} is reflexive transitive but not symmetric.