Finding Equivalence Relations in a Set of 4 Elements - Juan's Question

[galois26]

Our math Teacher asked us to find how many equivalence relations are there in a set of 4 elements, the set given is A={a,b,c,d} I found the solution to this problem there are 15 different ways to find an equivalence relation, but solving the problem, i looked in Internet that the number of equivalence relations (Partitions) of an n-element Set are the Bell numbers, somebody told me this is a definition and does not requiere a proof, but can this statement above be a theorem? If this is so I would like to see the proof.

Thanks in advance

Juan

 

[mathman]

The proof you need is not the name, but their specific values.
http://en.wikipedia.org/wiki/Bell_number