If R and S are relations on a set A, then prove the following :(i) R a

1.	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.
Answer» 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

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.

Answer»

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