# Interesting ways to show that there are infinitely many equivalence relations on an infinite set (including Bell numbers).

I am trying to answer the question "Is there infinitely many equivalence relations on any infinite set?"

My intuition says yes, and when I try to prove this, I feel like my reasoning is not sufficient. Here is what I have so far:

"The number of equivalence relations on an $$n$$-element set is given by the $$n$$th Bell number $$B_n$$, where

$$B_{n+1}=\sum\limits_{k=0}^{n} B_{k}{n\choose k}$$ for $$n\geq 0$$.

Now, the sequence of Bell numbers $$(B_n)_{n \in \mathbb{N}} \geq (n)_{n \in \mathbb{N}}$$ for all $$n \in \mathbb{N}$$. So I wanted to take limits (as n tends to infinity) across this inequality (which is allowed, by a basic result of analysis) and we see that $$\lim_{x \to +\infty} (B_n) = \infty$$ (since the limit of sequence of natural numbers is infinity). "

This "sort of" shows the result that I want, but it's not really explicitly showing it. Is there a final sentence I need to add to this to complete the proof, or is this method downright wrong? I feel like perhaps it is wrong as we're trying to give a result about infinite sets using finite objects.

I would also like to know if is there a more interesting way to prove this. Thanks

Here's a lower bound: Let $$X$$ be a set and $$A\subset X$$ be a subset of two or more elements. Then we can define an equivalence relation $$\sim_A$$ on $$X$$ per $$x\sim_A y\iff x=y\lor\{x,y\}\subseteq A.$$ Clearly, different $$A$$ lead to different equivalence relations. If $$X$$ is infinite, there are $$2^{|X|}$$ ways to pick $$A$$ and hence at least $$2^{|X|}$$ equivalence relations on $$X$$.
Let $$S$$ be any infinite set. Fix an element $$a\in S$$. Then for each $$b \in S$$, with $$b\ne a$$, define an equivalence relation, $$R_b$$ where one equivalence class is $$\{a,b\}$$, and all other equivalence classes are singletons. (So $$aR_b b, bR_b a$$, and everything else is only related to itself.)
Each $$R_b$$ is an equivalence relation, and the cardinality of $$\{R_b : b\in S, b\ne a\}$$ is equal to the cardinality of $$S$$ since $$S$$ is infinite.