The number of equivalence number of relation(Reflexive, Symmetric, Transitive):
The number of the equivalence relation is equal to partitions set into disjoint sets.
Case$(1):$ Suppose set is finite.Let it's cardinality is $n.$Then the number of partition is $B_{n}$.Where $B_{n}$ is Bell number determined by following recurrence:
$B_{n+1} =\sum_{k=0}^{n}\binom{n}{k}B_{k}$ $[$where $B_{0} =1]$
We have $B_{1}=1,B_{2} = 2,B_{3} =5,B_{4}=15,B_{5}=52$ and so on as per above recurrence.
Case$(2):$ Assume that cardinality of a set is infinite. In this case, the number of disjoint is partition is infinite. Thus the number of an equivalence class is infinite.
Resource of above concept is https://en.wikipedia.org/wiki/Bell_number
