Determine the intersection of equivalence relations

Question

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?

Answer 1

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 $\mathbb{Z}^2$.

Answer 2

In general, if $\sim$ is any equivalence relation on $\Bbb Z$ satisfying

$\tag 1 \text{For every } m \in \Bbb Z,\; m \sim (m+1)$

then for every $m, n \in \Bbb Z$. $\, m \sim n$.

We have $(x, 100-x) \in U$ and $(100-x,x+1) \in U$, so always $(x, x+1) \in R$. So $R = \Bbb Z \times \Bbb Z$.