[set theory] partition (partition | partition example | partition and equivalence relationship)

One 、 Divide

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Divide :

Nonempty set

A

A

A ,

A

≠

∅

A \not= \varnothing

A​=∅ ,

A

A

A One of the sets Divide yes Set family

A

\mathscr{A}

A ,
The Set family

A

\mathscr{A}

A Included in

A

A

A The power set of a set ,

A

⊆

P

(

A

)

\mathscr{A} \subseteq P(A)

A⊆P(A) , All elements in the set family belong to

A

A

A The power set of a set ;

Set family

A

\mathscr{A}

A The element in is aggregate , be called Partition ( Block ) , All the elements in the set are

A

A

A The elements in the collection ;

This set family

A

\mathscr{A}

A It has the following properties :

①

A

\mathscr{A}

A Every element in the set family is not empty

∅

∉

A

\varnothing \not\in \mathscr{A}

∅​∈A

②

A

\mathscr{A}

A Any two elements in the set family ( Partition / aggregate ) It's disjoint

∀

x

,

y

(

x

,

y

∈

A

∧

x

≠

y

⇒

x

∩

y

=

∅

)

\forall x,y ( x,y \in \mathscr{A} \land x \not= y \Rightarrow x \cap y = \varnothing )

∀x,y(x,y∈A∧x​=y⇒x∩y=∅)

③

A

\mathscr{A}

A Set all the elements in the family ( Partition / aggregate ) The union of is

A

A

A aggregate

⋃

A

=

A

\bigcup \mathscr{A} = A

⋃A=A

Quotient set is a division , The elements in this set family are set of equivalent classes ;

Quotient set reference : 【 Set theory 】 Equivalence class ( Concept of equivalence class | Examples of equivalence classes | Equivalence properties | Quotient set | Quotient set example ) Four 、 Quotient set

Two 、 Partition example

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The whole book is

E

E

E ,

take

E

E

E Of

n

n

n individual Extraordinary Of True subset , The meaning of non trivial is that it is neither an empty set , It's not itself ;

∅

≠

A

1

,

A

2

,

⋯
 

,

A

n

⊂

E

\varnothing \not= A_1 , A_2, \cdots, A_n \subset E

∅​=A1​,A2​,⋯,An​⊂E

1. Divide 1 be based on

1

1

1 Elements

Set family

A

i

=

{

A

i

,

∼

A

i

}

\mathscr{A}_i = \{ A_i , \sim A_i \}

Ai​={ Ai​,∼Ai​} ,

i

=

1

,

2

,

⋯
 

,

n

i = 1, 2, \cdots , n

i=1,2,⋯,n ,

A

i

\mathscr{A}_i

Ai​ The set family contains

A

i

A_i

Ai​ Set and its complement

∼

A

i

\sim A_i

∼Ai​ , This set family

A

i

\mathscr{A}_i

Ai​ Meet the three properties of the above division , It's a division ;

2. Divide 2 be based on

2

2

2 Elements

Set family

A

i

=

{

A

i

∩

A

j

,

∼

A

i

∩

A

j

,

A

i

∩

∼

A

j

,

∼

A

i

∩

∼

A

j

}

−

{

∅

}

\mathscr{A}_i = \{ A_i \cap A_j , \sim A_i \cap A_j , A_i \cap \sim A_j , \sim A_i \cap \sim A_j\} - \{ \varnothing \}

Ai​={ Ai​∩Aj​,∼Ai​∩Aj​,Ai​∩∼Aj​,∼Ai​∩∼Aj​}−{ ∅} ,

i

,

j

=

1

,

2

,

⋯
 

,

n

∧

i

≠

j

i,j = 1, 2, \cdots , n \land i \not= j

i,j=1,2,⋯,n∧i​=j

Understand according to the following Venn diagram :

• $A_i\cap A_j$
• $\sim A_i\cap A_j$
• $A_i\cap \sim A_j$
• $\sim A_i\cap \sim A_j$
• If $A_i$

3. Divide 3 be based on

3

3

3 Elements

Set family

A

i

j

k

=

{

A

i

∩

A

j

∩

A

k

,

A

i

∩

∼

A

j

∩

∼

A

k

,

∼

A

i

∩

A

j

∩

∼

A

k

,

∼

A

i

∩

∼

A

j

∩

A

k

,

∼

A

i

∩

∼

A

j

∩

∼

A

k

}

−

{

∅

}

\mathscr{A}_{ijk} = \{ A_i \cap A_j \cap A_k , A_i \cap \sim A_j \cap \sim A_k , \sim A_i \cap A_j \cap \sim A_k , \sim A_i \cap \sim A_j \cap A_k , \sim A_i \cap \sim A_j \cap \sim A_k\} - \{ \varnothing \}

Aijk​={ Ai​∩Aj​∩Ak​,Ai​∩∼Aj​∩∼Ak​,∼Ai​∩Aj​∩∼Ak​,∼Ai​∩∼Aj​∩Ak​,∼Ai​∩∼Aj​∩∼Ak​}−{ ∅}

4. Divide 4 be based on

n

n

n Elements

Set family

A

1

,

2

,

⋯
 

,

n

=

{

A

1

∩

A

2

∩

⋯

∩

A

n

,

A

1

∩

∼

A

2

∩

⋯

∩

∼

A

n

,

∼

A

1

∩

A

2

∩

⋯

∩

∼

A

n

,

⋮

∼

A

1

∩

∼

A

2

∩

⋯

∩

∼

A

n

}

−

{

∅

}

\begin{array}{lcl} \mathscr{A}_{1,2,\cdots,n} = \{ \\\\ A_1\cap A_2 \cap \cdots \cap A_n , \\\\ A_1\cap \sim A_2 \cap \cdots \cap \sim A_n , \\\\ \sim A_1\cap A_2 \cap \cdots \cap \sim A_n , \\\\ \vdots \\\\ \sim A_1\cap \sim A_2 \cap \cdots \cap \sim A_n \\\\ \} - \{ \varnothing \} \end{array}

A1,2,⋯,n​={ A1​∩A2​∩⋯∩An​,A1​∩∼A2​∩⋯∩∼An​,∼A1​∩A2​∩⋯∩∼An​,⋮∼A1​∩∼A2​∩⋯∩∼An​}−{ ∅}​

The rules :

A

1

A_1

A1​ To

A

n

A_n

An​ Union ,

n

n

n individual

∼

A

1

\sim A_1

∼A1​ To

∼

A

n

\sim A_n

∼An​ Union , Each of them is merged , Only one is not a complement ,

∼

A

1

\sim A_1

∼A1​ To

∼

A

n

\sim A_n

∼An​ Union ;

3、 ... and 、 Partition and equivalence theorem

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Partition and equivalence theorem :

Premise : aggregate

A

A

A Non empty ,

A

≠

∅

A \not= \varnothing

A​=∅

R

R

R The relationship is

A

A

A Equivalence relations on sets , Can be derived ,

A

A

A Set about

R

R

R Quotient set of relation

A

/

R

A/R

A/R yes

A

A

A Division ;

R

yes

A

On

etc.

price

Turn off

system

⇒

A

/

R

yes

A

Of

draw

branch

R yes A Superior price relationship \Rightarrow A/R yes A Division

R yes A On etc. price Turn off system ⇒A/R yes A Of draw branch

Set family

A

\mathscr{A}

A yes

A

A

A Partition on set , Define a Binary relationship yes Same block relationship

R

A

R_{\mathscr{A}}

RA​ ,
The Same block relationship yes

A

A

A On the assembly Equivalence relation ,
The Same block relationship yes Divided by

A

\mathscr{A}

A Defined relationships ;

x

R

A

y

⇔

∃

z

(

z

∈

A

∧

x

∈

z

∧

y

∈

z

)

xR_{\mathscr{A}}y \Leftrightarrow \exist z ( z \in \mathscr{A} \land x \in z \land y \in z )

xRA​y⇔∃z(z∈A∧x∈z∧y∈z)