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(B) An equivalence relation must be reflexive, symmetric, and transitive. $1$. Reflexivity: Since $R$ and $S$ are equivalence relations, $(a, a) \in R

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$R \cap S$ is an equivalence relation

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(B) An equivalence relation must be reflexive, symmetric, and transitive. $1$. Reflexivity: Since $R$ and $S$ are equivalence relations, $(a, a) \in R$ and $(a, a) \in S$ for all $a \in A$. Thus, $(a, a) \in R \cap S$. $2$. Symmetry: If $(a, b) \in R \cap S$, then $(a, b) \in R$ and $(a, b) \in S$. Since $R$ and $S$ are symmetric, $(b, a) \in R$ and $(b, a) \in S$, so $(b, a) \in R \cap S$. $3$. Transitivity: If $(a, b) \in R \cap S$ and $(b, c) \in R \cap S$, then $(a, b), (b, c) \in R$ and $(a, b), (b, c) \in S$. Since $R$ and $S$ are transitive, $(a, c) \in R$ and $(a, c) \in S$, so $(a, c) \in R \cap S$. Therefore, $R \cap S$ is an equivalence relation.

Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then

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