Question

If R and S are two equivalence relations on a set A, then R ∩ S is __________.

Answer 1

If R and S are two equivalence relations on a
Let A,
→ Since ∀ x∈A

(x, x)∈R and (x, x) ∈S ($\because$ R and S are reflexive)
⇒ (x, x) ∈ R ⋂ S ∀x∈A
i.e R ⋂ S is Reflexive

→ Let (x, y) $\in$R ⋂ S

⇒ (x, y) ∈R and (x, y) ∈S
⇒ (y, x)∈R and (y, x) ∈S ($\because$R and S are symmetric)
⇒ (y, x)∈R ⋂ S
⇒ R ⋂ S is symmetric

→ Let (x, y) (y, z) ∈ R ⋂ S

⇒ (x, y) (y, z) ∈ R and (x, y) (y, z) ∈ S as R and S are Transitive
⇒ (x, z) ∈ R and (x, z) ∈ S
⇒ (x, z)∈ R ⋂ S
i.e R ⋂ S is transitive

Hence, R ⋂ S is an equivalence relation