Determine the intersection of equivalence relations 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.
Let $$R:=\bigcap_{S\in S_U}S$$ which is an equivalence relation on $X$.
Suppose that $X=\mathbb{Z}$ and $U=\{(x,y)\in \mathbb{Z}^2\mid x+y\geq 100\}$, how can we define $R$ in that case?

Well, $R$ is always going to be an equivalence relation and we can use this to help us.
Note that $(x,|x|+k)\in U$ for any $x\in\mathbb{Z}$ and any $k\geq 100$. Hence, if we let $x,y\in \mathbb{Z},$ then $(x,|x|+|y|+100)\in U$ by the previous and, likewise, $(y,|x|+|y|+100)\in U.$ Accordingly, these elements are also in $R$. Since $R$ is an equivalence relation, it's transitive, so this implies $(x,y)\in R$.
We conclude that $R$ is the trivial relation $\mathbb{Z}^2$.
