What is a Quotient Set: Practical Terms & Euler's Phi Function

[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]

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?

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?

 

[chwie]

quotient set is the set of the equivalence class of a set X. You can think of it like if we can't distinguish between equivalent members and the quotient set is the set in which the whole set of equivalent members of a class is represented by an abstract member [a]. In Math is important the concept that:

quotient set<-> equivalence relations<-> partitions

The proof is standard.

About the phi functions there is no equivalence between co-primes in general, but like co-primes are defined using modules then the concept of equivalence is related. In number theory the congruence module something can be show to form a partition in the integers and also a quotient set.

 

[Landau]

...he said, three years later ;)

 

[chwie]

LOL

better late than never.