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