Question

Let R1 and R2 are two reflexive relations on a set A. Then which of the
following is correct?
(a) R1 U R2 is reflexive but R1 intersection R2 is not reflexive

(b) R1 intersection R2 is reflexive but R1 U R2 is not reflexive

(c) R1 U R2 and R1 intersection R2 both are reflexive

(d) Neither R U R2 nor R1 intersection R2 are reflexive​

Answer 1

Given:

Let R1 and R2 are two reflexive relations on a set A.

To find:

Which of the  following is correct?

Solution:

From the given statement, let us consider an example,

A = {1, 2, 3}

A × A = { (1,1) (2,2) (3,3) (1,2) (2,1) (1,3) (3,1) (2,3) (3,2) }

Reflexive Relation :- A relation R on a set A is said to be Reflexive if (x R x ) ∀ x ∈ A

Now Take any two relations on set A

R1 = { (1,1) , (2,2) , (3,3) , (1,2) }

R2 ={ (1,1) , (2,2) , (3,3) , (2,1) }

R1 ∪ R2 = { (1,1) , (2,2) , (3,3) , (1,2) , (2,1)  } which is Reflexive Relation.

R1 ∩ R2 = { (1,1) , (2,2) , (3,3)  } which is Reflexive Relation.

Therefore, option (c)   R1 U R2 and R1 intersection R2 both are reflexive is correct