Let R and S be equivalence relations on a set | vedclass

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(B) An equivalence relation on a set $A$ must satisfy three properties: reflexivity,symmetry,and transitivity. $1$. Reflexivity: Since $R$ and $S$ are

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$R \cap S$ is an equivalence relation on $A$.

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

Let $R$ and $S$ be equivalence relations on a set $A$. Then,

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