# Question about the defining equivalence relations on sets

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.

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 $$3. [e,f,g],[h,i]$$.