Equivalence Relations in A={a,b,c,d}: Proving the Bell Number Theorem

[lwpirelliwl]

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

 

[mathman]

What is given is the definition of Bell numbers. A proof is needed for the values as functions of n.

 

[Number Nine]

This exact question was posted verbatim in the General Math section. Methinks the OP has multiple accounts.

 

[lwpirelliwl]

No I have multiple accounts, I'm just wondering mate first and wanted to put the question in the right forum