How to calculate equivalence relations

Question

How can I calculate how many equivalence relations can be defined on a given set? For example:

How many possible equivalence relations can be defined on S = {a,b,c,d}?

Answer 1

Okay, so equivalence relations effectively partition $S$ into subsets where each element in a given subset is related to each other element in that subset. This means you simply need to count all the ways you can partition four elements.

Answer 2

Number of partitions of set is called Bell number and that number satissfy followin recurrence $$B_0=1,B_{n+1}=\sum_{k=0}^n {n\choose k}B_k$$ in your case $n=4$.