If `R_1` and `R_2` are equivalence relations in a set A, show that `R_1nnR_2` is also an equivalence relation.

Explanation

If we say `R₁` and `R₂` are two equivalence relations on a non-empty set `X`.
to prove that `R₁ R₂` in an equivalence relation on `X`

(1) `R₂` `R₂` is reflexive

Let `a in X` arbitrarily.

Then `(a, a) in R₁` and `(a, a) in R₂`
since `R₁`, `R₂` both being equivalence relations are reflexive.

So. `(a, a) in R₁ nn R₂`

⇒ `R₁` `R₂` is reflexive.

(ii) `R₁ nn R₂` is symmetric

Let `a, b = X` such that `(a, b) in R₁ nn R₂`
`(a, b) in R₁` and `(a, b) = R₂ = (b, a) = R₁` and `(b, a) in R₂`,
since `R₁` and `R₂` being equivalence relations are also symmetric.