Equivalence relations, equivalence classes and partitions

Question

While going through some discrete mathematics exercises I stumbled over the following statement:

there's a bijection between equivalence relations on a set S and the number of partitions on that set.

I KNOW

• the definition of "set, subset, equivalence relation, partition, equivalence class".

• further do I know what a bijection is.

• I was checking out on similar questions but I have not found any satisfying answers.

MY QUESTION

So every equivalence relation forms a disjoint partition of set S.

How do I find a bijection from any equivalence relation to the number of partitions on a set?

Furthermore I am puzzled with "the number of partitions". is the number of partitions the sum of all equivalence classes to every equivalence relation that exists on S?

And if my thoughts were correct, how do I create such an bijection?

I find it very confusing since one equivalence relation always produces I or more equivalence classes so I do not see how to create such an bijection.

WHY

I am solving an exercise and it is about this topic in fact its about finding out whether the number of all equivalence relations over set $\mathbb{N}$ is countable or not and prove that.

I do not want an answer to my question I just want to give you information for what I need help.

I would be very thankful for any helpful piece of advice and explanation which then would make it able for me to solve my exercise

Answer 1

I think you are mis-reading the statement. What it says is, there is a bijection from the set of equivalence relations to the set of partitions, which implies that the number of each is the same. Another way of saying the same thing, which is more consistent with the statement as given, is: Collect all partitions and assign a distinct number to each (partition #1, #2, ...); then there is a bijection between the set of equivalence relations and this set of numbers.

[So yeah, I think the statement is poorly phrased]

Answer 2

how do i create such an bijection?

Given an equivalence relation, you can make a partition into "mutually equivalent elements." Given a partition, you can declare two elements to be equivalent if they reside in the same piece of the partition.

furthermore i am puzzled with "the number of partitions".

An example: the partitions of $\{1,2,3\}$ are $\{\{1,2,3\}\}$, $\{\{1\},\{2,3\}\}$, $\{\{2\},\{1,3\}\}$, $\{\{3\},\{2,1\}\}$, and $\{\{1\},\{2\},\{3\}\}$. Five partitions.

is the number of partitions the sum of all equivalence classes to every equivalence relation that exists on S ?

You can't add equivalence classes in general, so no. You simply have to count how many ways the set can be split into disjoint subsets.