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

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The discussion revolves around finding the number of equivalence relations in a set of four elements, specifically the set A={a,b,c,d}. The original poster, Juan, states that there are 15 different equivalence relations and references the Bell numbers, which represent the number of partitions of an n-element set. A response clarifies that while the Bell numbers are indeed a definition, their specific values are what matter for solving the problem. The conversation emphasizes the need for understanding the proof behind the Bell numbers rather than just their name. Overall, the focus is on the relationship between equivalence relations and Bell numbers in the context of set theory.
galois26
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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

Juan
 
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