[set theory] set concept and relationship (set family | set family examples | multiple sets)

One 、 Set family

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Set family : except

P

(

A

)

P(A)

P(A) Beyond the power set , from Set of sets , It is called set family ;

Set family with index set : A set in a set family , Are marked , Is a family of sets with index sets ;

A

\mathscr{A}

A It is a family ,

S

S

S It's a collection

For any

α

∈

S

\alpha \in S

α∈S , There is Unique

A

α

∈

A

A_\alpha \in \mathscr{A}

Aα​∈A (

α

\alpha

α yes

S

S

S The elements in ,

A

α

A_\alpha

Aα​ Is a set family

A

\mathscr{A}

A Collection elements in )

also

A

\mathscr{A}

A Any set element in the set family , All corresponding

S

S

S An element in a collection

call

A

\mathscr{A}

A Set family In order to

S

S

S aggregate Set family of index set

S

S

S aggregate yes

A

\mathscr{A}

A Set family Of Index set

Write it down as :

A

=

{

A

α

∣

α

∈

S

}

\mathscr{A} = \{A_\alpha | \alpha \in S \}

A={ Aα​∣α∈S}

If you will

∅

\varnothing

∅ As a group ,

∅

\varnothing

∅ be called Empty set family ;

Two 、 Set family example

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1. Set family example 1 : The index set is limited , The set elements in the set family are finite

aggregate

A

1

=

{

1

}

A_1 = \{1\}

A1​={ 1} , aggregate

A

2

=

{

2

}

A_2 = \{ 2 \}

A2​={ 2} , that Set family

A

=

{

A

1

,

A

2

}

\mathscr{A} = \{ A_1 , A_2 \}

A={ A1​,A2​} In order to

{

1

,

2

}

\{1 , 2\}

{ 1,2} Set is the set of indicator set ;

2. Set family example 2 : The index set is limited , The set elements in the set family are finite

p

p

p Prime number

aggregate

A

k

=

{

x

∣

x

=

k

(

m

o

d

  

p

)

}

A_k = \{ x | x = k( mod \ \ p ) \}

Ak​={ x∣x=k(mod  p)} , among

k

=

0

,

1

,

2

,

⋯
 

,

p

−

1

k = 0, 1 , 2 , \cdots , p-1

k=0,1,2,⋯,p−1

Set family

A

=

{

A

0

,

A

1

,

A

2

,

⋯
 

,

A

p

−

1

}

\mathscr{A} = \{ A_0 , A_1 , A_2 , \cdots , A_{p-1} \}

A={ A0​,A1​,A2​,⋯,Ap−1​} In order to aggregate

{

0

,

1

,

2

,

⋯
 

,

p

−

1

}

\{0, 1 , 2 , \cdots , p-1\}

{ 0,1,2,⋯,p−1} For the index set Set family ;

Write it down as :

A

=

{

A

k

∣

k

∈

{

0

,

1

,

2

,

⋯
 

,

p

−

1

}

}

\mathscr{A} = \{ A_k | k \in \{0, 1 , 2 , \cdots , p-1\} \}

A={ Ak​∣k∈{ 0,1,2,⋯,p−1}}

3. Set family example 3 : The index set is infinite , The set elements in the set family are finite

aggregate

A

n

=

{

x

∈

N

 

∣

 

x

=

n

}

An = \{ x \in N \ | \ x = n \}

An={ x∈N ∣ x=n} It consists of a natural number element

n

n

n Set of components ;

Set family

A

=

{

A

n

∣

n

∈

N

}

\mathscr{A} = \{ A_n | n \in N \}

A={ An​∣n∈N} That is to say

N

N

N Set family of index set ;

4. Set family example 4 : Index set

N

+

N_+

N+​ Infinite , Every element in the set family, the elements in the set are also infinite ;

N

+

=

N

−

0

N_+ = N - {0}

N+​=N−0 ,

N

+

N_+

N+​ It's in addition to

0

0

0 Unexpected set of natural numbers

aggregate

A

n

=

{

x

 

∣

 

0

≤

x

<

1

/

n

∧

n

∈

N

}

A_n = \{ x \ | \ 0 \leq x < 1 / n \land n \in N \}

An​={ x ∣ 0≤x<1/n∧n∈N} ,

x

x

x yes

[

0

,

1

)

[0 , 1)

[0,1) Set of real numbers of intervals ,

n

n

n Express Division

0

0

0 Natural numbers other than ;

A

n

A_n

An​ The elements in a set are infinite , Its value range is

[

0

,

1

/

n

)

[ 0, 1/n )

[0,1/n) , It's an interval ;

Set family

A

=

{

A

n

∣

n

∈

N

+

}

\mathscr{A} = \{ A_n | n \in N_+ \}

A={ An​∣n∈N+​} That is to say

N

+

N_+

N+​ Set family of index set ;

3、 ... and 、 Multiple sets

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Multiple sets : The complete

E

E

E ,

E

E

E The elements in , Gather many times

A

A

A It appears that , call aggregate

A

A

A Is a multiple set ;

Repeatability :

E

E

E The elements in

a

a

a stay aggregate

A

A

A in appear

k

k

k Time , call

a

a

a The elements are in

A

A

A The repeatability in the set is

k

k

k ;

Examples of multiple sets :

The complete

E

=

{

a

,

b

,

c

,

d

}

E = \{a, b, c, d \}

E={ a,b,c,d}

Multiple sets

A

=

{

a

,

a

,

a

,

c

,

c

,

d

}

A = \{ a , a , a , c , c , d \}

A={ a,a,a,c,c,d} ,

a

a

a The elements are in

A

A

A The repeatability of the set is

3

3

3

b

b

b The elements are in

A

A

A The repeatability of the set is

0

0

0

c

c

c The elements are in

A

A

A The repeatability of the set is

2

2

2

d

d

d The elements are in

A

A

A The repeatability of the set is

1

1

1

The relationship between sets and multiple sets : The set can be regarded as the repetition degree is less than or equal to

1

1

1 Multiple sets of ;