Meeting

In this item, I explain a meeting in the mathematics. Please see "a meeting" (the Koran) about the sura of the Koran.

Speaking roughly, a meeting (a meeting British: set, Buddha: ensemble, Germany: Menge) in the mathematics is "the meeting" consisting of "somethings". Former (げん British: element; element) is individual "things" constituting a meeting.

It is one of the most basic concepts in the whole modern mathematics, and, as for the meeting, it may be said that most of the modern mathematics are written by words of a meeting and the representation as well as set theory.

A certain meeting may be conventionally called a system (Kei British: system) or a family (ぞく British: family). There is no essential difference in these names, but it is really thought that I include the difference in small nuance. For example, pro-equation (set of the equation "uniting mutually") interrelated groups (set of the meeting based on "the constant rule"), an addition group (interrelated groups "having an additive property").

Table of contents

Introduction

A meeting is "a meeting" of "the things". The target "thing" collected as former (element) of the meeting does not mind anything including a number, a letter, a sign (but of course it is a meeting).

On the other hand, you may not call any "meeting" a meeting. An object "can be decided without an uncertain element uniquely whether it is the cause of the meeting" and must be defined ように so that the "meeting" is called a meeting.

For example, whole soot {♠,♦,♣,♥} of cards and whole number {A, 2 of cards, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K} are examples of the meetings. The cards are these groups (except a joker)

{(♠,A), ..., (♠,K), (♦,A), ..., (♦,K), (♣,A), ..., (♣,K), (♥,A), ..., (♥,K)}

It is 52 pieces of cards to assume を secret language, but this also becomes an example of the meeting. This in particular is an example of the prompt shipment meeting with a set of the soot and the set of the numbers, and 52 expresses the density of this meeting again. In addition, the density of set of the former soot, set of the number is 4, 13 each.

Sign of cards

	A	2	3	4	5	6	7	8	9	10	J	Q	K
♠	(♠,A)	(♠,2)	(♠,3)	(♠,4)	(♠,5)	(♠,6)	(♠,7)	(♠,8)	(♠,9)	(♠,10)	(♠,J)	(♠,Q)	(♠,K)
♦	(♦,A)	(♦,2)	(♦,3)	(♦,4)	(♦,5)	(♦,6)	(♦,7)	(♦,8)	(♦,9)	(♦,10)	(♦,J)	(♦,Q)	(♦,K)
♣	(♣,A)	(♣,2)	(♣,3)	(♣,4)	(♣,5)	(♣,6)	(♣,7)	(♣,8)	(♣,9)	(♣,10)	(♣,J)	(♣,Q)	(♣,K)
♥	(♥,A)	(♥,2)	(♥,3)	(♥,4)	(♥,5)	(♥,6)	(♥,7)	(♥,8)	(♥,9)	(♥,10)	(♥,J)	(♥,Q)	(♥,K)

Capital letter A, B, of the Roman letters that it is often it to place holder expressing a meeting ..., E, F, ..., M, N, ..., S, T, ..., X, Y, ... Using など [¹], the material (corresponding to the letter which I used to express a meeting in particular) of the meeting is Latin small letter a, ..., e, ..., m, ..., s, ..., x, ... There is much とすることが [²].

Reversion and inclusion

"Former" (mathematics) and "membership" are referred to for the details "inclusion relations" "a subset"

 

Inclusion relations: A is a part of B. B is on A.

I can think about the simple relations that I include it between a meeting and the cause, a meeting and the meetings and am included in.

Membership

When target a is one of the things constituting meeting A, it is said with "meeting A having a as an element" that "a is an element of meeting A" (or the cause) and expresses which "a belongs to meeting A" with a ∈ A or A ∋ a.

Inclusion relations

When I consist when all the causes to belong to A belong to B about two meeting A, B without, in other words, x ∈ A ⇒ x ∈ B depending on how to get a, it is said, "B includes A", and which "A is a subset of B" writes down which "A is included in B as a meeting" with A ⊂ B or A ⊆ B or B ⊃ A or B ⊇ A.

Membership and the inclusion relations are different concepts and must not confuse it. For example, it is X ⊂ Z, but X ∈ Z is not necessarily led by all means from X ∈ Y ∈ Z if it is X ⊂ Y ⊂ Z. In addition, if it is x ∈ A ⊂ B, it is x ∈ B, but it is not possible to come to the conclusion in x ∈ B generally from x ⊂ A ∈ B.

Scale

The notation of the meeting includes two ways of methods roughly. A logical concept includes what's called "broadening and connotation", but I am almost equivalent to it, and am met by all means if included in a method to enumerate all the elements and the meeting and am a method to exhibit the condition that is not met by all means if not included.

By the method to enumerate all equivalent to "broadening", for example, 1, 3, 5, 7, the meeting consisting of 9,

${\displaystyle 1,3,5,7,9}$ 

I write と.

A meeting of the whole equilateral odd number, for example, less than 10 by the method to exhibit the condition that you should meet to belong equivalent to "connotation,"

{x | Regular odd number} that x is less than 10

I write と. Generally, the meeting that gathered all only the objects which satisfied it when there was condition P(x),

${\displaystyle x\mid P(x)}$ 

I write と. Use a variable called x here; {y | Even if write it as P(y)}; {a | You may write it as P(a)}. I say the connotation notation in set-builder notation (en:Set-builder notation) and set comprehension, Japanese. It comes from broadening and the connotation of the logical concept like the said article each, and those words are well still used in the field of mathematics in the Japanese zone, but do not see extension and intension which are each original in the English zone too much now in this field.

Condition P(x) is {x by a meeting to be decided [³] that I am often given in form that "x is the material of X and satisfies condition Q(x) more" then | Instead of writing it like x ∈ X and Q(x)}; often easily

${\displaystyle x\in X\mid Q(x)}$ 

I sketch などと. Meeting {x ∈ X | Q(x)} becomes the subset of X. In addition, condition P(x) is meeting {x at the time of form "that I can express with x = f(y) using a certain y satisfying condition Q(y)" | P(x)}

${\displaystyle f(y)\mid Q(y)}$ 

I may express のように.

A set of the whole meeting that cannot write all about an element extensively, e.g., natural number

${\displaystyle 0,1,2,3,\dots}$ 

May write のように, but "...Because there is room to produce misunderstanding as for the truncation by ", such a scale is limited when the meaning of the abbreviated contents is clear.

Axiom of the extensionality

When I assume A, B any meeting, if "X is an element of A, X is an element of B only for time," but does it about any meeting X if I consist when A and B are equal. In other words,

${\displaystyle \forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\Rightarrow A=B)}$ 

である.

I refer to "the axiom of the extensionality"

As intuitive explanation, for example, is {1, 3, 5, 7, 9} and {x | Regular odd number} that x is less than 10 is different expression, but is that it is an equal meeting because it is a meeting to assume natural number 1, 3, 5, 7, 9 an element both.

Special meeting

In the mathematics, I think about the meeting having no factor toward. Because there is merely such a meeting only in one according to the principle of the extensionality, I say an empty set (British: empty set) and express this in ∅. ∅ is a subset of any meeting A. This is because x ∈ ∅ ⇒ x ∈ A is the truth than x∉∅ for any target x. There are some sets expressed by the sign that, besides, was decided of the empty set:

• ${\displaystyle \mathbb {N}}$  I express a set of the whole は natural number.
• ${\displaystyle \mathbb {Z}}$  I express a set of the whole は integer.
• ${\displaystyle \mathbb {Q}}$  I express a set of the whole は rational number.
• ${\displaystyle \mathbb {R}}$  I express a set of the whole は real number.
• ${\displaystyle \mathbb {C}}$  I express a set of the whole は complex number.
• ${\displaystyle \mathbb {H}}$  I express a set of は 四元数全体.
• ${\displaystyle \mathbb {U}}$  I express は グロタンディーク space.

Density

The details refer to "density" (mathematics)

Call the set consisting of the causes of the limited unit a limited meeting (ゆうげんしゅうごう British: finite set); the original number of meeting A #(A), |A| I often express it in signs such as, card(A). The set that is not a limited meeting is called an infinite meeting (むげんしゅうごう British: infinite set). I widen a concept of "the number" for the infinite meeting and think about what's called density (のうど British: potency) or radix (I come and breathe it British: cardinal number, British: cardinality). Instead of counting the number, I label a different meeting with the cause (British: indexing; additional character charge account) and, using a certain meeting, do it and check whether one-on-one correspondence is produced. Then I can confirm that expansion of "the number" to an infinity meeting and the concept that it is are decided properly because the density of limited meeting is fixed by the just original number.

All the infinite meetings have the cause of "the infinite unit", but is which infinity fir tree; is not same, and distinguish the infinity that there is many it in the concept of the density, and will treat it. For example, the fact to have the density that real number is different from the natural number that natural number and rational number have the same density toward truly is the contents which were well known for a person learning mathematics. In a similar fact, plane R² and number line R have the same density, and it is spoken that several kinds of mysterious plane curves called the plane filling curve covering up a plane exhaustively exist. It is similar in a more dimensional high space, and space filling curve to fill up the space is built. As for space with a different dimension having the same density, a dimension and density express that one is the different standard that does not virtually measure the other.

Operation of the meeting

When I handle some meetings and discuss it about the relationship, what originally I make a new meeting from meetings that I thought about and check is one of the effective means. These operation provides some algebraic systems about interrelated groups by considering it to be the operation for the meeting. I will offer some concepts again if I consider those algebraic systems to be an abstract algebraic system by applying generalization of the abstract algebra.

Basic meeting operation

End, merger, the sum

The details refer to "a merger" (set theory)

 

Schematic view of the end

"I attach" two meetings and can take out the meeting that て is what I make together, and is new. One of the operation becoming basic of additive interrelated groups. Sum of sets。

${\displaystyle A\cup B:= x\mid x\in A\lor x\in B}.$ 

Acquaintance, crossing-over, the product

The details refer to "crossing-over" (set theory)

 

Schematic view of the acquaintance

I can take out a new meeting by finding the common part of two meetings. The operation that becomes basic of multiplication-like interrelated groups. Common part。

${\displaystyle A\cap B:= x\mid x\in A\land x\in B}.$ 

Naokazu, 非交和

The details refer to "非交和"

The sum not to have the acquaintance of two meetings.

Assistant difference, aspect pair

The details refer to "difference of two sets"

 

Schematic view of the difference of two sets

About one meeting of two meetings, is included in the other among the causes to belong to it at the same time; get the money's worth, and remove it, and can make a new meeting. The difference is one and an acquaintance with the other complement and is multiplication-like operation.

${\displaystyle A\smallsetminus B:= x\mid x\in A\land x\notin B}.$ 

Assistant supplementary absolute

The details refer to "difference of two sets"

 

Schematic view of the complement

A difference from the whole of the meeting one under the supposition that a universal set (universal class) is given, and any meeting is a subset of the universal sets. The selfish meeting does not have the complement and acquaintance, and those sums accord in a universal set.

${\displaystyle \complement A:= x\mid x\notin A}.$ 

Boolean sum

The details refer to "a Boolean sum"

 

Schematic view of the Boolean sum

From the cause to belong to the end of two meetings, belong to the acquaintance; get the money's worth, and remove it, and can think about a new meeting. This is the difference that attracted an acquaintance from an end. The operation that is additive like an end.

${\displaystyle A\,\triangle \,B:=(A\smallsetminus B)\cup (B\smallsetminus A).}$ 

The instructions function gives means to replace these meeting operation with world algebraic operation consisting of 0 and 1.

${\displaystyle A\leftrightarrow \mathbf {1} _{A}(x):={\begin{cases}1&(x\in A)\\0&(x\notin A)\end{cases}}.}$ 

Other operation

One universal set is given the operation mentioned above, and the operation result virtually becomes the subset of the again same universal set if sets to become the argument of the operation are the subsets. On the other hand, there is the operation that it cannot necessarily expect.

冪

The details refer to "a power set"

 

Schematic view of 冪 of the 3 yuan meeting

For a given meeting, the power set is a set of the whole meeting included in the given meeting. Even if the power set of a certain meeting says that I am greatest in the interrelated groups consisting of the subsets of the meeting, it is the same.

${\displaystyle {\mathcal {P}}(X):= S\mid S\subseteq X}.$ 

Prompt shipment

The details refer to "a prompt shipment set"

 

Schematic view of the prompt shipment

For two meetings, I can make a meeting to assume an original prioritized pair to belong to each an element.

${\displaystyle X\times Y:= (x,y)\mid x\in X\land y\in Y}.$ 

Placement meeting, representation space

The details refer to "a placement set"

The meeting that, as a whole, is new is found if I consider the representation from a certain meeting to a different meeting to be one cause. I can see the prompt shipment meeting with a placement set consisting of the representation to let any meeting support each cause of the ordinal number.

${\displaystyle Y^{X}:= f\mid f{\text{ is a mapping from }}X{\text{ to }}Y}.$ 

Quotient

The details refer to "a quotient set"

 

Schematic view of the classification of the meeting

When I give a meeting a classification, I can think about a meeting to assume each kind the element.

${\displaystyle X/{\sim} curly brace = [x]\mid x\in X ,{\text{ where }}[x]:= y\in X\mid y\sim x}.$ 

Some interrelated groups

The details refer to "the algebra of the meeting" and "interrelated groups"

I think about family A consisting of meetings. When A meets some properties about meeting operation, they may be given the special name.

• Of [⁴] that is a multiplication group when pro-π-(English version) π-system includes an empty set when A is closed about crossing-over (limited) [⁵] (the ディンキン group refers, too). Furthermore, it is said that it is a δ-multiplication group when it is closed about countable crossing-over. In addition, for any two meetings that a multiplication group has inclusion relations toward, it is said with a meeting and a half ring when I have a line to perform 非交和 in a limitedness time from one, and to reach the other.
• When A is closed about crossing-over with the sum (limited) (limited), I say the bunch of the meeting or a ring. I may call only for the case that A is not an empty set, and is closed about (or includes an empty set as the cause), the sum and a difference (or it is the same thing, but is closed about a Boolean sum and crossing-over) with a meeting ring. Furthermore, it is said with a σ-meeting ring if closed about countable crossing-over if I close it about a δ-meeting ring, the countable sum. In addition, I say algebra or a body if these include a universal set. δ-aggregate is σ-aggregate.
• A includes an empty set (limited); the sum and supplementary; it is said that is the shut じているとき addition group that is particularly a limited addition group. Furthermore, I say a complete addition group if closed about the countable sum. Interrelated groups A is equivalent with it being aggregate to be an addition group, and a complete addition group is another name for σ-aggregate likewise.
• The interrelated groups that the monotonous family is closed about the limit of the monotonous line about inclusion relations
• ディンキン族（d-族、δ-族）は全体集合を含み、包含関係を持つ集合同士の差について閉じていて、可算増大列の極限について閉じている。λ-系は全体集合を含み、補について閉じていて、可算非交和について閉じている。この二つは同じ概念を定める。
• 層族はそれに属する任意の集合 A, B が A ⊂ B または A ⊃ B または A ∩ B ≠ ∅ の何れか一つのみを満たす。
• ブール環

注記

1. ^ 定数や変数に対する慣例を踏襲して A, B, ... や X, Y, ... が使われるほか、英語の set, ドイツ語の Menge, フランス語の ensemble の頭文字 S, M, E やその周辺の文字がよく使われる。
2. ^ ラテンアルファベット以外にもギリシャ文字を使うこともある。集合の集合を考えるときは、元である集合に大文字を使うことから、筆記体 ${\displaystyle \scriptstyle {\mathcal {A,\ldots ,E,\ldots ,M,\ldots ,S,\ldots ,X,\ldots }}}$ やドイツ文字 ${\displaystyle \scriptstyle {\mathfrak {A,\ldots ,M,\ldots ,X,\ldots }}}$ で記したりする。このような入れ子構造は何重にも複雑な形で現われたり、同じものが違った見方をされたりするので、このような文字種の変更を行わないこともよくある。
3. ^ 「x が X の元であって」というような断り書きをしない場合にも、実際には「普遍集合」 (英: universal set) あるいは「宇宙」 (英: universe) と呼ばれる、必要な議論を展開することができる程度に十分大きな集合を考え、集合と言えば必ずその普遍集合の部分集合だけを考えているといったようなことがしばしば行われる。条件 P(x) の形から x の属するべき集合 X がある程度限定される場合にも、断り書きはしばしば省略される。
4. ^ 例えば定義 2.1.
5. ^ しばしば π-系と乗法族はこれと逆に扱われたり同義語の場合もある。例えば定義 1.3.6.や[1]は乗法族 (multiplicative class) に交叉について閉じていることのみを課している。

関連項目

ウィクショナリーに集合の項目があります。

• 写像
• 集合論 - 素朴集合論/公理的集合論
• 定義 - 外延と内包
• セット (抽象データ型) - コンピュータ上での実装

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