The division of the meeting

Figure (expression with the Euler's diagram) which divided a meeting into six parts

In mathematics, I say that I divide the whole X into the partial / block / cell which is not piled up each other with division (partition) of meeting X. Judging from a divided meeting, those "cells" become exclusive, complete universal set (MECE) mutually formally.

Table of contents

Definition

The division of meeting X is a meeting of the subsets that is not empty of X, and individual former x of X belongs only to one subset by all means.

Because set P of the meeting that is not empty is the division of X, it is necessary for next to be managed.

1. The original sum of sets of P is equal to X (the cause of P covers X).
2. The common part of any two different causes of P is an empty set (as for the cause that, in other words, varies in P in each other bare である).

I can express these two conditions as follows mathematically.

1. ${\displaystyle \bigcup P=X}$ 
2. ${\displaystyle A\cap B=\varnothing {\text{ if }}A\in P,\,B\in P,\,A\neq B}$ 

I call the cause of P "block (block)" of the division or "partial (part)" [¹].

Example

• There is only only division {{x}} in every singleton {x}.
• About any meeting X which is not empty, P = {X} is one of the division of X.
• About any truth subset A which is not empty of meeting U, A and the difference of two sets are one of the division of U.
• Meeting {1, 2, 3} have five kinds of division.
  • I may transcribe {{1}, {2}, {3} of} this into 1/2/3.
  • I may transcribe {{1, 2}, {3} of} this into 12/3.
  • I may transcribe {{1, 3}, {2} of} this into 13/2.
  • I may transcribe {{1}, {2, 3} of} this into 1/23.
  • I may transcribe {{1, 2, 3} of} this into 123.

By the way

• {curly brace, {1,3}, {2}} are not the division (to include an empty set).
• {{1,2}, {2, 3}} are not the division (because the cause of 2 is included in two subsets).
• {{1}, {2}} are not {1, 2, 3} division (because 3 is included in no block), but are right for {1, 2} division.

The division and equivalence relation

About any equivalence relation R on meeting X, set X/R of the equivalence class is the division of X. I call that I get the division of X as the equivalence class meeting classification of X by R or classification (classification) from equivalence relation R of meeting X [²].

On the contrary, I can define equivalence relation R_P on X from any division P of X. In other words, this establishes equivalence relation if I assume it x ~ y when any two former x and y of X belong to the same block of P. This time, equivalence relation R_P called the (associated) equivalence relation associated with division P [²].

Therefore, the division of the meeting is equivalent with setting equivalence relation for a meeting essentially [³].

Subdivision of the division

When division π of meeting X is subdivision (refinement) of division ρ of meeting X, I say that all the individual causes of π are original subsets of one of ρ. Speaking roughly, the division is more careful π than p. I may transcribe this into π≤ρ.

These "more careful" relations in the set of the division of X are half order (suitable therefore expressing it in "≤") and, actually, are a perfection bunch. For example, X = {1, 2, 3, 4} "division bunch" have 15 yuan and are expressed in a figure of following Hasse.

 

I speak a method to subdivide the division from the viewpoint of equivalence relation as another example. I assume D the set of 52 pieces of cards of general cards. I transcribe relations, "a color is the same" in D into ~_C. Two equivalence class, meeting called {red card} and meeting called {black card} are provided then. There is subdivision by ~_S, "soot is the same" concerned in the division of 2 blocks corresponding to this ~_C, and four equivalence class {spade}, {diamond}, {heart}, {club} are provided.

The division that is a non-intersection

Set N = {1 of the natural number, 2, ...When when the division corresponding to equivalence relation ~ of, n} is non-intersection (noncrossing), number a, b, c, d according to each in N are big things and small things relations called a <b <c <d, and, furthermore, there is not a thing called a ~ c and b ~ d. With X = {1 mentioned above, 2, 3, 4}, only 13/24 is the division that is not a non-intersection. In late years the bunch of the division that was the non-intersection of the limited meeting understood that it was important in free probability theory. Because operation to take the end of two bunches is not equal, these form the subset of the bunch of all division, but are not a partial bunch.

Of various division count it up, and it

The total number of division of the set having the material of the n unit is Bell number B_n. It becomes B0 = 1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, B6 = 203 when I enumerate small Bell number of n. The Bell number is expressed in next 漸化式.

${\displaystyle B_{n+1}=\sum _{k=0}^{n}{n \choose k}B_{k}}$ 

And following index type generating function exists.

${\displaystyle \sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}z^{n}=e^{e^{z}-1}}$ 

It is number of the kind sterling S(n, k) second as for the total number of the division to divide the set having the material of the n unit into the block of the k unit.

The total number of the division that is the non-intersection of the set having the material of the n unit is Catalan number C_n and is expressed in the next expression.

${\displaystyle C_{n}={1 \over n+1}{2n \choose n}}$ 

Allied item

• Data cluster ring
• Equivalence relation

Footnote, source

1. ^ Brualdi, Richard A. (2004). Introductory Combinatorics (4th edition ed.). Pearson Prentice Hall. pp. 44-45. ISBN 0131001191. 
2. ^ ^{a b} Kazuo Matsuzaka "meeting, phase guide" Iwanami Shoten, 1968. p. 57
3. ^ Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. p. 54. ISBN 0126227608. 

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