Question

Let R and S be two equivalence relations on set A. Prove that $R\cap S$ is an equivalence relation.

Answer 1

Equivalence relation mean reflexive, symmetric and transitive.
Let an element $a\in A$. Since R and S are equivalence relations they are reflexive.
Therefore $(a,a)\in R$ and $(a,a)\in S$
So $(a,a)\in R$ and $(a,a)\in S$
So $(a,a)\in R\cap S$
$\therefore R\cap S$ is reflexive ......(1)
Let $(a,b)\in R\cap S$
$\Rightarrow (a,b)\in R$ and $(a,b)\in S$
$\Rightarrow (b,a)\in R$ and $(b,a)\in S$
( Since R and S are symmetric),
$\Rightarrow (b,a)\in R\cap S$
$\therefore R\cap S$ is symmetric ....(2)
Let $(a,b),(b,c)\in R\cap S$
$\Rightarrow (a,b),(b,c)\in R\Rightarrow (a,c)\in R$
$\Rightarrow (a,b),(b,c)\in S\Rightarrow (a,c)\in S$
Since R and S are transitive,
$\Rightarrow (a,c)\in R\cap S$
$\Rightarrow R\cap S$ is transitive ....(3)
From (1), (2) and (3), $R\cap S$ is an equivalence relation.