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

Explanation

Suppose that `R_{1}` and `R_{2}` are two equivalence relations on a non-empty set `X`.


First we prove that `{R}_{1} cap {R}_{2}` in an equivalence relation on `{X}`.


(i) `R_{2} cap R_{2}` is reflexive :


Let `a in X` arbitrarily.


Then `(a, a) in R_{1}` and `(a, a) in R_{2}`, since `R_{1}, R_{2}` both being equivalence relations are reflexive.


So. `(a, a) in R_{1} cap R_{2}`


`Rightarrow R_{1} cap R_{2}` is reflexive.


(ii) `R_{1} cap R_{2}` is symmetric :


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