# If R and S are relations on a set A, then prove the following :(i) R and S are symmetric ⇔ R ⋂ S, and R ⋃ S is symmetric(ii) R is reflexive, and S is any relation ⇔ R ⋃ S is reflexive.

R and S are two symmetric relations on set A

(i) To prove: R S is symmetric

Symmetric: For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let (a, b) R S

(a, b) R and (a, b) S

(b, a) R and (b, a) S

[ R and S are symmetric]

(b, a) R S

R S is symmetric

To prove: R S is symmetric

Symmetric: For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let (a, b) R S

(a, b) R or (a, b) S

(b, a) R or (b, a) S

[ R and S are symmetric]

(b, a) R S

R S is symmetric

(ii) R and S are two relations on a such that R is reflexive.

To prove : R S is reflexive

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Suppose R S is not reflexive.

This means that there is a R S such that (a, a) R S

Since a R S,

a R or a S

If a R, then (a, a) R

[ R is reflexive]

(a, a) R S

Hence, R S is reflexive

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