Question about the defining equivalence relations on sets

Question

Suppose I have an equivalence relation $\sim$ on $S=\{e,f,g,h,i\}$ such that $e \sim f, f \sim g$ and $e \nsim i$. I’m trying to find the number of such relations that can be defined on $S$. I know that $\{e,f,g\}$ will always be an equivalence class and that $\{i\}$ will also always be an equivalence class. The questions therefore is equivalent to asking how many different equivalence classes can $h$ belong to and the answer is obviously $3$ since it can belong to its own equivalence class $\{h\}$, $\{i\}$ or $\{e,f,g\}$. However I’m not sure if it’s possible that $h$ does not belong to any equivalence class, i.e. the set of equivalence classes for the relations would be $\{\{e,f,g\},\{i\}\}$. I think the answer is no because the set of equivalence classes has to partition $S$ but I’m not 100% sure.

Answer 1

An equivalence relation most certainly has to completely partition the set. It follows from the reflexivity requirement: at the very least, each element must be equivalent to itself, therefore constituting an equivalence class of its own.

Answer 2

There is a bijection between the set of equivalence relations on a set, and the partitions of that set. This allows us to switch between these notions fluidly.

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That is, we know that $e,f,g$ are in the same equivalence class, which is one that is different from $i$. So, it is a question of where you place $h$, as you mentioned. The answer to your last part , is that it has to be part of its own equivalence class, since it is related to itself, and therefore must appear in one of the classes. Alternately, a partition covers every element, so must cover $h$. This gives us just the three possibilities, of $1.[e,f,g,h],[i]$ , $2 . [e,f,g],[h],[i]$, and $3. [e,f,g],[h,i]$.