[set theory] set concept and relationship (set represents | number set | set relationship | contains | equality | set relationship property)

One 、 Set theory system

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Set theory system :

• Naive set theory : Inclusion paradox ; Naive set theory in Cannot define a set precisely ;
• Axiomatic set theory : In order to eliminate the paradox in naive set theory , Axiomatic set theory established ; Axiomatic set theory is more rigorous , Describe what a set is by a set of axioms ;

Two 、 Sets represent

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Sets represent : Use Capital Represents a collection , Lowercase letters Represents an element in a collection ;

Enumeration : List all the elements in the set , Use commas to separate elements , Use curly braces “{}” Cover up ; Such as :

A

=

{

0

,

1

,

2

,

3

}

A = \{0, 1, 2, 3\}

A={ 0,1,2,3} ,

B

=

{

0

,

1

,

2

,

3

,

⋯
 

}

B = \{0, 1, 2, 3, \cdots\}

B={ 0,1,2,3,⋯}

Description : Use The predicate

P

(

x

)

P(x)

P(x) Express

x

x

x Have the quality of

P

P

P , Use

{

x

∣

P

(

x

)

}

\{x | P(x)\}

{ x∣P(x)} Indicates that it has properties

P

P

P Set ;

P

(

x

)

P(x)

P(x) Express

x

x

x It's English letters ,

{

x

∣

P

(

x

)

}

\{ x | P(x) \}

{ x∣P(x)} Represents the English alphabet ;

P

(

x

)

P(x)

P(x) Express

x

x

x It's even ,

{

x

∣

P

(

x

)

}

\{ x | P(x) \}

{ x∣P(x)} Represents an even set ;

Sets represent considerations :

No repetition : Collection There can be no repeating elements ;

lack order : The elements in the set are A disorderly ;

Set representation transformation : The representations of sets can be transformed into each other , Description and Enumeration Can transform each other ;

Example of representation transformation :

Enumeration :

A

=

{

0

,

2

,

4

,

6

,

⋯
 

}

A=\{ 0, 2, 4 , 6 , \cdots \}

A={ 0,2,4,6,⋯}

Description :

A

=

{

x

∣

x

≥

0

and

And

x

yes

accidentally

Count

}

A = \{ x | x \geq 0 also x It's even \}

A={ x∣x≥0 and And x yes accidentally Count }

3、 ... and 、 Count the sets

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Set of natural numbers :

N

=

{

0

,

1

,

2

,

⋯
 

}

N = \{ 0, 1 , 2 , \cdots \}

N={ 0,1,2,⋯}

Set of integers :

Z

=

{

0

,

±

1

,

±

2

,

⋯
 

}

Z = \{ 0, \pm 1 , \pm 2 , \cdots \}

Z={ 0,±1,±2,⋯}

The set of rational numbers :

Q

Q

Q

Set of real numbers :

R

R

R

Complex sets :

C

C

C

3、 ... and 、 A collection of relations

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A collection of relations Yes Inclusion relation , Equality relation , In addition, the nature of the relationship is Self reflection , Antisymmetry , Transitivity ;

1、 Inclusion relation

The containment relationship of the set :

describe :

A

,

B

A, B

A,B Two sets , If

B

B

B The elements in All are

A

A

A The elements in , call

B

B

B aggregate yes

A

A

A A collection of A subset of ,

A

A

A contain

B

B

B ,

B

B

B Included in

A

A

A ;

Write it down as :

B

⊆

A

B \subseteq A

B⊆A

Symbolic form :

B

⊆

A

⇔

∀

x

(

x

∈

B

→

x

∈

A

)

B \subseteq A \Leftrightarrow \forall x ( x \in B \to x \in A )

B⊆A⇔∀x(x∈B→x∈A) , For all objects , As long as it belongs to

B

B

B aggregate , Belong to

A

A

A aggregate ;

The set does not contain relationships :

describe : If aggregate

B

B

B No aggregate

A

A

A Subset

Write it down as :

B

⊈

A

B \not\subseteq A

B​⊆A ;

Symbolic form :

B

⊈

A

⇔

∃

x

(

x

∈

B

∧

x

∉

A

)

B \not\subseteq A \Leftrightarrow \exist x ( x \in B \land x \not\in A )

B​⊆A⇔∃x(x∈B∧x​∈A) , For all objects , The existing object belongs to

B

B

B aggregate , Do not belong to

A

A

A aggregate ;

Include examples :

A

=

1

,

2

,

3

,

4

A = {1, 2, 3, 4}

A=1,2,3,4 ,

B

=

1

,

2

,

3

B = {1, 2, 3}

B=1,2,3 ,

C

=

1

,

2

C = {1, 2}

C=1,2

Yes

C

⊆

B

C \subseteq B

C⊆B ,

C

⊆

A

C \subseteq A

C⊆A ,

B

⊆

A

B \subseteq A

B⊆A

2、 Equality relation

The equality of sets :

describe :

A

,

B

A, B

A,B Two sets , If

A

A

A contain

B

B

B , also

B

B

B contain

A

A

A , said

A

A

A And

B

B

B equal ;

Write it down as :

A

=

B

A = B

A=B

Symbolize :

A

=

B

⇔

∀

x

(

x

∈

B

x

∈

A

)

A = B \Leftrightarrow \forall x ( x \in B \leftrightarrow x \in A )

A=B⇔∀x(x∈Bx∈A)

3、 The property of containing relation between sets

The property of containing relation between sets : Below

A

,

B

,

C

A, B, C

A,B,C It's three sets , The following propositions are true propositions ;

reflexivity :

A

⊆

A

A \subseteq A

A⊆A , The set really contains itself ;

Antisymmetry : if

A

⊆

B

A \subseteq B

A⊆B And

B

≠

A

B \not= A

B​=A , be

B

⊈

A

B \not\subseteq A

B​⊆A
( This property is equivalent to if

A

⊆

B

A \subseteq B

A⊆B And

B

⊆

A

B \subseteq A

B⊆A , be

A

=

B

A = B

A=B )

Transitivity : if

A

⊆

B

A \subseteq B

A⊆B And

B

⊆

C

B \subseteq C

B⊆C , be

A

⊆

C

A \subseteq C

A⊆C