Quotient sets商集合しょうしゅうごう and the canonical projection自然な射影

1Introduction

After classifying objects対象たいしょう by an equivalence relation同値関係どうちかんけい, the next question is whether the resulting boxes themselves can be treated as elements元げん. This idea is the quotient set商集合しょうしゅうごう.

In a quotient set, individual elements are replaced by equivalence classes同値類どうちるい as the new objects. In other words, it forgets fine distinctions and collapses equivalent objects to the same point点てん.

2Terms and definitions

For an equivalence relation同値関係どうちかんけい $$\sim$$\sim on a set集合しゅうごう $$A$$A, define the quotient set商集合しょうしゅうごう $$A/\sim$$A/{\sim} by

Here $$\left[a\right]$$[a] is the equivalence class同値類どうちるい of $$a$$a.

Order note: the formal lecture on maps写像しゃぞう appears later. On this page, $$\pi :A\to A/\sim$$\pi:A\to A/{\sim} is used only in the minimal sense of a rule assigning $$\left[a\right]$$[a] to each $$a\in A$$a\in A.

The canonical projection自然な射影 $$\pi :A\to A/\sim$$\pi:A\to A/{\sim} is defined by

It is the map写像しゃぞう sending each element元げん to the equivalence class containing it.

3Strategy

When working with a quotient set商集合しょうしゅうごう, distinguish a representative代表元だいひょうげん from an equivalence class同値類どうちるい. The object $$a$$a is an element元げん of $$A$$A, while $$\left[a\right]$$[a] is an element of $$A/\sim$$A/{\sim}.

When defining a map写像しゃぞう on a quotient set, check well-definednesswell-defined 性. Even if the same equivalence class is written using a different representative, the defined value値あたい must not change.

As a procedure, first check that the relation is an equivalence relation同値関係どうちかんけい, and then form the equivalence classes同値類どうちるい. The canonical projection自然な射影 sends each element元げん to the equivalence class containing it.

4Intuitive explanation

A quotient set商集合しょうしゅうごう is the set of the boxes produced by classification. The canonical projection自然な射影 sends each element元げん to “the box it belongs to.”

For example, if integers are classified by remainders modulo $$3$$3, the quotient set is

Here $$\left[1\right]=\left[4\right]=[-2]$$[1]=[4]=[-2]. The representatives代表元だいひょうげん differ, but the equivalence class同値類どうちるい they denote is the same.

It is useful to think of an equivalence class同値類どうちるい as a box. One may compute using a chosen representative代表元だいひょうげん, but one must check that the result does not depend on which representative was chosen; this is the check of well-definednesswell-defined 性.

5What is well-definednesswell-defined 性?

In a quotient set商集合しょうしゅうごう, one element元げん can be written using several representatives代表元だいひょうげん. Suppose a rule assigns a value $$f\left(a\right)$$f(a) in a set $$B$$B to each $$a\in A$$a\in A, and we want to define $$\overline{f}:A/\sim \to B$$\bar f:A/{\sim}\to B by $$\overline{f}\left(\right[a\left]\right)=f\left(a\right)$$\bar f([a])=f(a). For this candidate rule to be independent of the representative, we must check that $$a\sim b$$a\sim b implies $$f\left(a\right)=f\left(b\right)$$f(a)=f(b).

This check is the check of well-definednesswell-defined 性. If well-definedness fails, the same element of the quotient set can receive different values値あたい depending on the chosen representative.

To prove well-definednesswell-defined 性, take two equivalent representatives $$a,b$$a,b and prove $$f\left(a\right)=f\left(b\right)$$f(a)=f(b). Computing with only one representative is not enough.

6Worked example: the canonical projection自然な射影

6.1Problem

On $$\mathbb{Z}$$\mathbb Z, define $$a\sim b$$a\sim b to mean that $$a-b$$a-b is a multiple of $$3$$3. Find $$\pi \left(5\right)$$\pi(5) and $$\pi (-1)$$\pi(-1) for the canonical projection自然な射影 $$\pi :\mathbb{Z}\to \mathbb{Z}/\sim$$\pi:\mathbb Z\to\mathbb Z/{\sim}.

6.2Explanation

The canonical projection自然な射影 is $$\pi \left(a\right)=\left[a\right]$$\pi(a)=[a]. Therefore

Since $$5\sim 2$$5\sim2 and $$5\sim -1$$5\sim -1, this is the same equivalence class同値類どうちるい as $$\left[2\right]$$[2] and $$[-1]$$[-1].

Also

and $$[-1]=\left[2\right]=\left[5\right]$$[-1]=[2]=[5]. Therefore $$\pi \left(5\right)=\pi (-1)$$\pi(5)=\pi(-1). This example shows that the canonical projection collapses equivalent elements元げん to the same equivalence class.

7How to identify it and related links

• If boxes formed by an equivalence relation同値関係どうちかんけい are treated as elements元げん, think of a quotient set商集合しょうしゅうごう.
• Distinguish $$a$$a from $$\left[a\right]$$[a]. The former is an element of the original set集合しゅうごう, while the latter is an element of the quotient set.
• If a map写像しゃぞう on a quotient set is defined using representatives, check well-definednesswell-defined 性.
• The canonical projection自然な射影 sends each element to its equivalence class同値類どうちるい.

8Exercise links