If R and S are two equivalence relations on a set A,
Question:

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

Solution:

If R and S are two equivalence relations on a

Let A,

→ Since ∀ xA

$(x, x) \in R \quad$ and $\quad(x, x) \in S(\because R$ and $S$ are reflexive $)$

$\Rightarrow(x, x) \in R \cap S \quad \forall x \in A$

i.e $R \cap S$ is Reflexive

$\rightarrow$ Let $(x, y) \in R \cap S$

$\Rightarrow(x, y) \in R$ and $(x, y) \in S$

$\Rightarrow(y, x) \in R$ and $(y, x) \in S(\because R$ and $S$ are symmetric $)$

$\Rightarrow(y, x) \in R \cap S$

$\Rightarrow R \cap S$ is symmetric

$\rightarrow \operatorname{Let}(x, y)(y, z) \in R \cap S$

$\Rightarrow(x, y)(y, z) \in R$ and $(x, y)(y, z) \in S$ as $R$ and $S$ are Transitive

$\Rightarrow(x, z) \in R$ and $(x, z) \in S$

$\Rightarrow(x, z) \in R \cap S$

i.e   R ⋂ S is transitive

Hence, R ⋂ S is an equivalence relation