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

 

[blue_raver22]

Yes, the statement above can be considered a theorem known as the Bell Number Theorem. It states that the number of equivalence relations (or partitions) on a set of n elements is equal to the nth Bell number. This can be proven using combinatorial arguments, induction, or generating functions.

One possible proof is as follows:

First, let's define the Bell numbers. The Bell numbers, denoted as B(n), are a sequence of numbers that count the number of ways to partition a set of n elements. For example, B(3) = 5 because there are 5 ways to partition a set of 3 elements: {a,b,c}, {a, {b,c}}, {b, {a,c}}, {c, {a,b}}, and {a,b,c}.

Now, let's consider a set A with n elements. We want to find the number of equivalence relations on this set.

First, let's consider the case where n = 1. In this case, there is only one equivalence relation, which is the identity relation. Therefore, B(1) = 1.

Next, let's consider the case where n = 2. In this case, there are 3 possible equivalence relations: {a,b}, {a, {b}}, and {b, {a}}. Therefore, B(2) = 3.

Now, let's assume that for some k > 2, the statement holds true: the number of equivalence relations on a set of k elements is equal to the kth Bell number, B(k).

We want to show that this holds true for n = k+1.

Consider a set A with k+1 elements. Let's choose one element, say a, from this set. Now, we can partition the remaining k elements in B(k) ways. For each of these partitions, we can include a in either one of the existing subsets or create a new subset containing only a. Therefore, for each of the B(k) partitions, we have 2 options for including a. This gives us a total of 2*B(k) ways to partition a set of k+1 elements.

However, we also need to consider the case where a is in its own subset, which is already included in the B(k) partitions we counted earlier. Therefore, we need to subtract B(k) from the total to avoid double counting.