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