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

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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
 
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What is given is the definition of Bell numbers. A proof is needed for the values as functions of n.
 
This exact question was posted verbatim in the General Math section. Methinks the OP has multiple accounts.
 
No I have multiple accounts, I'm just wondering mate first and wanted to put the question in the right forum
 

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