Let $X$ be a set. We consider the relations on $X$ as subsets of $X\times X$. Let $U\subseteq X\times X$ be a subset, and let $S_U$ be the set of all equivalence relations on $X$ that contain $U$ as subset.
Show that $$R:=\bigcap_{S\in S_U}S$$is an equivalence relation on $X$.
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For that we have to show that $R$ is reflexive, symmetric and transitive.
We have that $S_U$ is the set of all equivalence relations on $X$, therefore an element $S\in S_U$ is an equivalence relation, isn't it?
An element of $R$ is of the form $S_1\cap S_2\cap \ldots \cap S_n$, where $S_i\in S_U$ are equivalence relations.
So we have to show that the intersections of equivalence relations are still equivalence relations, or not?
I think you're having a bit trouble with the notation. You are correct in thinking that an element $S \in S_U$ is an equivalence relation. Each $S \in S_U$ is an equivalence relation and therefore some subset of $X \times X$. So $R$, which is the intersection of all such $S$'s is also a subset of $X \times X$. So an element of $R$ is of the form $(x,y) \in X \times X$, and by the construction of $R$ we know that $(x,y) \in S$ for any $S$ which is an equivalence relation om $X$.
Now that we've untangled the definitions/notations, you should be able to see how to finish the problem.