How many different equivalence relations with exactly two different equivalence classes are there on a set with $n$ elements I came across with this topic. It looks straight forward for $5$ elements, but what if I want to find how many different equivalence relations with exactly two different equivalence classes are there on a set with $n$ elements?
It says that the set of equivalence relations on a set are in direct bijection with the set of partitions on a set. This made me read about Bell numbers but I'm not sure how help us here. How to solve this problem?

Pick an element $a\in S$. Then we have an obvious bijection between $$\{\text{equivalence relations on }S\text{ with exactly two equivalence classes}\}$$ and $$\{\text{proper subsets of }S\setminus\{a\}\,\}$$
given by
$$ {\sim}\mapsto \{\,x\in S\setminus\{a\}\mid x\sim a\,\}$$
