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

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Discussion Overview

The discussion centers on the concept of quotient sets, particularly in relation to equivalence relations and their practical implications. Participants explore how quotient sets partition a set into equivalence classes and examine potential connections to Euler's phi function.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants seek a practical understanding of quotient sets, noting that they partition a larger set into discrete subsets.
  • One participant explains that a quotient set is derived from an equivalence relation, representing equivalent elements as a single class.
  • Another participant asserts that surjective functions on a set are equivalent to equivalence relations, partitions, and quotient sets of that set.
  • A participant mentions that any total map from a set A to a set B induces a partition of A, leading to a corresponding quotient set.
  • There is a suggestion that the partition of the positive integers (Z+) induced by Euler's phi function may have relevance in analyzing the properties of phi.
  • One participant emphasizes the relationship between quotient sets, equivalence relations, and partitions, stating that this connection is fundamental in mathematics.
  • Another participant notes that while there is no general equivalence between co-primes, the concept of equivalence can be related to modular arithmetic, which forms partitions in the integers.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and interpretation of quotient sets and their relation to Euler's phi function. No consensus is reached on the specific applications of quotient sets in relation to phi, and multiple viewpoints are presented regarding the nature of equivalence in number theory.

Contextual Notes

Some discussions involve assumptions about the definitions of equivalence relations and partitions, which may not be universally agreed upon. The relationship between quotient sets and Euler's phi function remains speculative and is not fully resolved.

lordy12
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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?
 
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lordy12 said:
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.
 
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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.
 
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?
 
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.
 
...he said, three years later ;)
 
LOL

better late than never.
 

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