quotient set

lordy12

what exactly is a quotient set? I know it "partitions" a large group of numbers into discrete subsets but I still don't know what exactly it is in practical terms. Like, does it relate somehow to Euler's phi function?

Hurkyl

Suppose you have a set and an equivalence relation on it. Intuitively, a quotient set is what you get when you make equivalent things equal.

In set theory, the "standard" quotient set is the set of equivalence classes. In other words, the "standard" way to make equivalent things equal is to replace everything with its equivalence class.

mathwonk

surjective functions defined on S are equivlent to equivalence relations on S and equivalent to partitions of S and equivalent to quotient sets of S.

fopc

A not empty.
Any (total) map f:A->B determines a partition of its domain in an obvious way. If R is the associated equivalence relation on A, then the partition is the quotient set A/R. The members of A/R are equivalence classes.

I know next to nothing about phi, but it looks like
dom(phi) = Z+.
Is the partition of Z+ (induced by phi) used anywhere in the rather lengthy analysis of phi properties?