当前位置：网站首页>[set theory] order relation (partial order relation | partial order set | example of partial order set)

[set theory] order relation (partial order relation | partial order set | example of partial order set)

2022-07-03 07:22:00 【Programmer community】

List of articles

One 、 Partial order relation
Two 、 Posets
3、 ... and 、 Examples of partial order relationships ( Greater than or equal to 、 Less than or equal to 、 to be divisible by | Ordered pairs of elements are single values )
Four 、 Examples of partial order relationships 2 ( Inclusion relation | An ordered pair of elements is a set )
5、 ... and 、 Examples of partial order relationships 3 ( Refinement relation | Order is a family of elements )

One 、 Partial order relation

Partial order relation :

Given a non empty set

$A$ ,

≠

∅

A \not= \varnothing

$A \neq = \emptyset$ ,

$R$ The relationship is

$A$ Binary relations on sets ,

⊆

R \subseteq A \times A

$R \subseteq A \times A$ ,
If

$R$ The relationship satisfies the following properties :

introspect : All vertices in the graph They all have rings ;
antisymmetric : Between two vertices Yes
$1$ A directed edge ;
Pass on : Premise $\to b , b\to ca→b,b→c Don't set up Default delivery ; Premise a → b , b → c a \to b , b\to ca→b,b→c establish Must satisfy a → c a \to ca→c There is ;$

said

$R$ The relationship is

$A$ On the assembly Partial order relation ;

Partial order relation representation : Use

≼

\preccurlyeq

$≼$ Symbols indicate partial order relations , pronounce as “ Less than or equal to ” ;

Symbolize :

∈

⇔

≼

<x,y> \in R \Leftrightarrow xRy \Leftrightarrow x \preccurlyeq y

$< x, y > \in R \Leftrightarrow x R y \Leftrightarrow x ≼ y$ , Reading :

<x,y>

$< x, y >$ Ordered pairs are in partial order relation

$R$ in , be

$x$ And

$y$ There are

$R$ Relationship ,

$x$ Less than or equal to

$y$ ;

Equivalence relation Is used for classification Of , Partial order relation Is used for organization Of , Inside each class , Give a structure ;

Two 、 Posets

Posets :

≼

\preccurlyeq

$≼$ Relationship yes

$A$ Partial order relations on sets , said aggregate

$A$ And Partial order relation

≼

\preccurlyeq

$≼$ Composed of Ordered pair

≼

<A, \preccurlyeq>

$< A, ≼ >$ It is called a poset ;

If there is a partial order relation on a set , Then this set is called a poset ;

3、 ... and 、 Examples of partial order relationships ( Greater than or equal to 、 Less than or equal to 、 to be divisible by | Ordered pairs of elements are single values )

Greater than or equal to 、 Less than or equal to 、 Division relations yes Ordered pair set , Each of them Elements of ordered pairs yes Single numeric element ;

1. Greater than or equal to 、 Less than or equal to

∅

≠

⊆

\varnothing \not=A \subseteq R

$\emptyset \neq = A \subseteq R$ , Nonempty set

$A$ , Is a set of real numbers

$R$ Subset ;

aggregate

$A$ The greater than or equal relationship on , Less than or equal to , Are partial order relations , Both relationships satisfy introspect , antisymmetric , Pass on Relationship ;

The poset is expressed as :

≤

≥

$< A, \leq >, < A, \geq >$

The greater than or equal relationship set represents :

≥

{

∣

∈

∧

≥

}

\geq = \{<x, y>\ | x,y \in A \land x \geq y \}

$\geq = {< x, y > ∣ x, y \in A \land x \geq y}$

The less than or equal relationship set represents :

≤

{

∣

∈

∧

≤

}

\leq = \{<x, y>\ | x,y \in A \land x \leq y \}

$\leq = {< x, y > ∣ x, y \in A \land x \leq y}$

2. Division relations

∅

≠

⊆

{

∣

∈

∧

}

\varnothing \not=A \subseteq Z_+ = \{ x | x \in Z \land x > 0 \}

$\emptyset \neq = A \subseteq Z_{+} = {x ∣ x \in Z \land x > 0}$ , Nonempty set

$A$ , Is a set of positive integers

Z_+

$Z_{+}$ Subset ;

aggregate

$A$ Upper Division relations It's a partial order relationship , Divisible relations are all satisfied introspect , antisymmetric , Pass on Relationship ;

The poset is expressed as :

∣

<A , |>

$< A, ∣ >$

The set of divisible relations represents :

∣

{

∣

∈

∧

∣

}

|= \{<x, y>\ | x,y \in A \land x | y \}

$∣ = {< x, y > ∣ x, y \in A \land x ∣ y}$

x
x
$x$ to be divisible by
y
y
$y$ ,
x
∣
y
x|y
$x ∣ y$ ,
x
x
$x$ It's a divisor ( molecular ) ,
y
y
$y$ It's a dividend ( The denominator ) ;
y
x
\dfrac{y}{x}
$\frac{y}{x}$
Reference resources : 【 Set theory 】 Binary relationship ( Special relationship types | Empty relation | Identity | Global relations | Division relations | Size relationship ) 3、 ... and 、 Division relations

Four 、 Examples of partial order relationships 2 ( Inclusion relation | An ordered pair of elements is a set )

Inclusion relation yes Ordered pair set , Each of them Elements of ordered pairs yes aggregate ;

Set family

\mathscr{A}

$A$ Included in

$A$ The power set of a set ,

⊆

(

)

\mathscr{A}\subseteq P(A)

$A \subseteq P (A)$ ;
Inclusion relation ,

$x$ Included in

$y$ , Symbolize :

⊆

{

∣

∈

∧

⊆

}

\subseteq = \{<x,y> | x,y \in \mathscr{A} \land x \subseteq y \}

$\subseteq = {< x, y > ∣ x, y \in A \land x \subseteq y}$

Include examples of relationships :

Premise :

aggregate
A
=
{
a
,
b
}
A = \{ a, b \}
$A = {a, b}$
Set family
A
1
=
{
∅
,
{
a
}
,
{
b
}
}
\mathscr{A}_1 = \{ \varnothing , \{ a \} , \{ b \} \}
$A_{1} = {\emptyset, {a}, {b}}$
Set family
A
2
=
{
{
a
}
,
{
a
,
b
}
}
\mathscr{A}_2 = \{ \{ a \} , \{ a , b \} \}
$A_{2} = {{a}, {a, b}}$
Set family
A
3
=
P
(
A
)
=
{
∅
,
{
a
}
,
{
b
}
,
{
a
,
b
}
}
\mathscr{A}_3 = P(A) = \{ \varnothing , \{ a \} , \{ b \} , \{ a , b \} \}
$A_{3} = P (A) = {\emptyset, {a}, {b}, {a, b}}$ , This family is also
A
A
$A$ Power set of ;

Use ordered pairs to represent the following containment relationships , Every element of an ordered pair is a set ;

① Set family

\mathscr{A}_1

$A_{1}$ All inclusive relationships on :

⊆

∪

{

∅

{

}

∅

{

}

\subseteq_1 = I_{\mathscr{A}_1} \cup \{ <\varnothing , \{ a \}> , <\varnothing , \{ b \}> \}

$\subseteq_{1} = I_{A_{1}} \cup {< \emptyset, {a} >, < \emptyset, {b} >}$

The identity relation on a set family is an inclusion relation , Any set is contained in the set itself ;
An empty set is contained in any non empty set ;

② Set family

\mathscr{A}_2

$A_{2}$ All inclusive relationships on :

⊆

∪

{

}

{

}

\subseteq_2 = I_{\mathscr{A}_2} \cup \{ <\{ a \}, \{ a, b \}> \}

$\subseteq_{2} = I_{A_{2}} \cup {< {a}, {a, b} >}$

The identity relation on a set family is an inclusion relation , Any set is contained in the set itself ;

{

}

\{ a \}

${a}$ Set contained in

{

}

\{ a, b \}

${a, b}$ aggregate ;

③ Set family

\mathscr{A}_3

$A_{3}$ All inclusive relationships on :

⊆

∪

{

∅

{

}

∅

{

}

∅

{

}

{

}

{

}

{

}

{

}

\subseteq_3 = I_{\mathscr{A}_3} \cup \{ <\varnothing , \{ a \}> , <\varnothing , \{ b \}> , <\varnothing , \{ a,b \}> , \{ <\{ a \}, \{ a, b \}> \} , \{ <\{ b \}, \{ a, b \}> \} \}

$\subseteq_{3} = I_{A_{3}} \cup {< \emptyset, {a} >, < \emptyset, {b} >, < \emptyset, {a, b} >, {< {a}, {a, b} >}, {< {b}, {a, b} >}}$

The identity relation on a set family is an inclusion relation ;
An empty set is contained in any non empty set ;

{

}

\{ a \}

${a}$ Set contained in

{

}

\{ a, b \}

${a, b}$ aggregate ;

{

}

\{ b \}

${b}$ Set contained in

{

}

\{ a, b \}

${a, b}$ aggregate ;

5、 ... and 、 Examples of partial order relationships 3 ( Refinement relation | Order is a family of elements )

Refinement relation yes Ordered pair set , Each of them Elements of ordered pairs yes Set family ;

aggregate

$A$ Non empty ,

\pi

$π$ yes

$A$ Set divided into sets , Every partition is a set family ;

Divide reference : 【 Set theory 】 Divide ( Divide | Partition example | Partition and equivalence )

There is a relationship between sets , Refinement relation , Using symbols

≼

Add

fine

\preccurlyeq_{ Refine }

$≼_{Add fine}$ Express ;

Refinement relation

≼

Add

fine

\preccurlyeq_{ Refine }

$≼_{Add fine}$ Symbolize :

≼

Add

fine

{

∣

∈

∧

yes

Add

fine

}

\preccurlyeq_{ Refine } = \{ <x, y> | x, y \in \pi \land x yes y The refinement of \}

$≼_{Add fine} = {< x, y > ∣ x, y \in π \land x yes y Of Add fine}$

Premise :

aggregate
A
=
{
a
,
b
,
c
}
A = \{ a, b , c \}
$A = {a, b, c}$
Set family
A
1
=
{
{
a
,
b
,
c
}
}
\mathscr{A}_1 = \{ \{ a , b , c \} \}
$A_{1} = {{a, b, c}}$
Set family
A
2
=
{
{
a
}
,
{
b
,
c
}
}
\mathscr{A}_2 = \{ \{ a \} , \{ b , c \} \}
$A_{2} = {{a}, {b, c}}$
Set family
A
3
=
{
{
b
}
,
{
a
,
c
}
}
\mathscr{A}_3= \{ \{ b \} , \{ a , c \} \}
$A_{3} = {{b}, {a, c}}$
Set family
A
4
=
{
{
c
}
,
{
a
,
b
}
}
\mathscr{A}_4= \{ \{ c \} , \{ a , b \} \}
$A_{4} = {{c}, {a, b}}$
Set family
A
5
=
{
{
a
}
,
{
b
}
,
{
c
}
}
\mathscr{A}_5= \{ \{ a \} , \{ b \} , \{ c \} \}
$A_{5} = {{a}, {b}, {c}}$

The above families are

$A$ Partition of sets ;

Divide the set , Form a new set ;

{

}

\pi_1 = \{ \mathscr{A}_1 , \mathscr{A}_2 \}

$π_{1} = {A_{1}, A_{2}}$

{

}

\pi_2 = \{ \mathscr{A}_2 , \mathscr{A}_3 \}

$π_{2} = {A_{2}, A_{3}}$

{

}

\pi_3 = \{ \mathscr{A}_1 , \mathscr{A}_2 , \mathscr{A}_3 , \mathscr{A}_4, \mathscr{A}_5 \}

$π_{3} = {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}}$

①

\pi_1

$π_{1}$ The refinement relationship of partition elements in the set :

≼

∪

{

}

\preccurlyeq_1 = I_{\pi1} \cup \{<\mathscr{A}_2, \mathscr{A}_1>\}

$≼_{1} = I_{π 1} \cup {< A_{2}, A_{1} >}$

Each division is its own refinement ;

\mathscr{A}_2

$A_{2}$ yes

\mathscr{A}_1

$A_{1}$ The refinement of ,

\mathscr{A}_2

$A_{2}$ Less than or equal to

\mathscr{A}_1

$A_{1}$ ;

②

\pi_2

$π_{2}$ The refinement relationship of partition elements in the set :

≼

\preccurlyeq_2 = I_{\pi2}

$≼_{2} = I_{π 2}$

Each division is its own refinement ;

\mathscr{A}_2 , \mathscr{A}_3

$A_{2}, A_{3}$ Neither of them is the refinement of the other ;

③

\pi_3

$π_{3}$ The refinement relationship of partition elements in the set :

≼

∪

{

}

\preccurlyeq_3 = I_{\pi3} \cup \{<\mathscr{A}_2, \mathscr{A}_1> , <\mathscr{A}_3, \mathscr{A}_1> , <\mathscr{A}_4, \mathscr{A}_1> , <\mathscr{A}_5, \mathscr{A}_1> , <\mathscr{A}_5, \mathscr{A}_2> , <\mathscr{A}_5, \mathscr{A}_3> , <\mathscr{A}_5, \mathscr{A}_4> \}

$≼_{3} = I_{π 3} \cup {< A_{2}, A_{1} >, < A_{3}, A_{1} >, < A_{4}, A_{1} >, < A_{5}, A_{1} >, < A_{5}, A_{2} >, < A_{5}, A_{3} >, < A_{5}, A_{4} >}$

Each division is its own refinement ;
Any division is

\mathscr{A}_1

$A_{1}$ The refinement of ;

\mathscr{A}_1

$A_{1}$ It's the biggest , Greater than or equal to any other partition ;

\mathscr{A}_5

$A_{5}$ It is the refinement of any division ;

\mathscr{A}_5

$A_{5}$ The smallest is the smallest , Less than or equal to any other partition ;

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当前位置：网站首页>[set theory] order relation (partial order relation | partial order set | example of partial order set)

[set theory] order relation (partial order relation | partial order set | example of partial order set)

List of articles

One 、 Partial order relation

Two 、 Posets

3、 ... and 、 Examples of partial order relationships ( Greater than or equal to 、 Less than or equal to 、 to be divisible by | Ordered pairs of elements are single values )

Four 、 Examples of partial order relationships 2 ( Inclusion relation | An ordered pair of elements is a set )

5、 ... and 、 Examples of partial order relationships 3 ( Refinement relation | Order is a family of elements )

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