Let $X$ be a set s.t $|X|=5$ , How many equivalence relations are there from X to itself?
I tried to do it "manually" and I tried to create all possible relationships,
Considering $X$ has Cardinality of 5 we can write that $X=${$a_i|i\in ${$1,..,5$} }
And I tried to list all the possible equivalence relationships, but I find it inelegant, and considering the fact that I previously proven that there's a bijection between the set of all partitions and the set of all equivalence relationships, it would be enough to enumerate all the possible partitions, but I have no idea how to construct all the possible partitions, I have defined the partition as cover of non-void disjunctive sets, and based from this definition I don't know how to proceed