[set theory] order relation (total order relation | total order set | total order relation example | quasi order relation | quasi order relation theorem | bifurcation | quasi linear order relation | q

One 、 The whole sequence ( Line order relation )

---

A

A

A Set and Partial order relation

≼

\preccurlyeq

≼ The ordered pair of composition is :

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> Posets ;

A

A

A Collection Any element

x

,

y

x, y

x,y all Comparable ;

said

≼

\preccurlyeq

≼ The relationship is

A

A

A On the assembly The whole sequence , Also known as Line order relation ;

call

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> Is a totally ordered set ( Line ordered set ) ;

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> Posets Is a totally ordered set

If and only if

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> Hastu of a poset is a straight line

Two 、 Examples of total order relationships

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Nonempty set

A

A

A Included in Set of real numbers

R

R

R ,

∅

≠

A

⊆

R

\varnothing \not= A \subseteq R

∅​=A⊆R ,

A

A

A On the assembly Greater than or equal to

≥

\geq

≥ , Less than or equal to

≤

\leq

≤ All are

A

A

A On the assembly The whole sequence ,

<

A

,

≤

>

<A , \leq>

<A,≤> ,

<

A

,

≥

>

<A , \geq>

<A,≥> yes Totally ordered sets ;

Hastu is a straight line ;

3、 ... and 、 Coherent relations

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Nonempty set

A

A

A , Binary relationship

R

R

R yes

A

A

A Binary relations on sets ;

Symbolize :

A

≠

∅

A \not= \varnothing

A​=∅ ,

R

⊆

A

×

A

R \subseteq A \times A

R⊆A×A ;

If Binary relationship

R

R

R yes Reflexion , Pass on Of ,

said

R

R

R The relationship is

A

A

A Quasi ordered relations on sets ,

Use

≺

\prec

≺ Denotes a quasi ordered relation ,

call

<

A

,

≺

>

<A , \prec>

<A,≺> Is a quasi ordered set ;

Partial order relation

≼

\preccurlyeq

≼ yes Less than or equal to Relationship , Coherent relations

≺

\prec

≺ Namely Strictly less than Relationship ;

Examples of quasi ordered Relations : Greater than , Less than , It really includes , Are all quasi ordered Relations ;

Coherent relations The complete nature is Reflexion , antisymmetric , Pass on ,
The reason why it is not mentioned in the concept antisymmetric nature , Because according to Reflexion , Transitive nature , Can be derived antisymmetric nature ;

Mathematics tends to use the minimum condition to define , Therefore, antisymmetry is removed here ;

Four 、 Quasi ordered relation theorem 1

---

Nonempty set

A

A

A ,

A

≠

∅

A \not= \varnothing

A​=∅ ,

≼

\preccurlyeq

≼ Yes no empty set

A

A

A The partial order relations on the ,

≺

\prec

≺ Yes no empty set

A

A

A Quasi ordered relation on ;

① Properties of partial order relation :

≼

\preccurlyeq

≼ yes introspect , antisymmetric , Delivered

② Properties of quasi ordered relation :

≺

\prec

≺ yes Reflexion , antisymmetric , Delivered

③ Partial order relation -> Coherent relations : Partial order relation subtract Identity Namely Coherent relations ,

≼

−

I

A

=

≺

\preccurlyeq - I_A = \prec

≼−IA​=≺

④ Coherent relations -> Partial order relation : Coherent relations And Identity The union of is Partial order relation ,

≺

∪

I

A

=

≼

\prec \cup I_A = \preccurlyeq

≺∪IA​=≼ ;

Four 、 Quasi ordered relation theorem 2

---

Nonempty set

A

A

A ,

A

≠

∅

A \not= \varnothing

A​=∅ ,

≺

\prec

≺ Yes no empty set

A

A

A Quasi ordered relation on ;

①

x

≺

y

x \prec y

x≺y ,

x

=

y

x=y

x=y ,

y

≺

x

y \prec x

y≺x At most one of them is established ;

Use counter evidence , Any two of them will lead to

x

≺

x

x \prec x

x≺x ;

②

(

x

≺

y

∧

x

=

y

)

∧

(

y

≺

x

∧

x

=

y

)

⇒

x

=

y

(x\prec y \land x = y) \land (y \prec x \land x=y) \Rightarrow x = y

(x≺y∧x=y)∧(y≺x∧x=y)⇒x=y

5、 ... and 、 Trisomy 、 Quasilinear sequence

---

Nonempty set

A

A

A ,

A

≠

∅

A \not= \varnothing

A​=∅ ,

≺

\prec

≺ Yes no empty set

A

A

A Quasi ordered relation on ;

If

x

≺

y

x \prec y

x≺y ,

x

=

y

x=y

x=y ,

y

≺

x

y \prec x

y≺x There is only one city , So called

≺

\prec

≺ Coherent relations have Trisomy ;

There are three differences Reverse order relation

≺

\prec

≺ be called

A

A

A On the assembly Quasilinear order relation , Also known as quasi total order relation ;

<

A

≺

>

<A \prec>

<A≺> go by the name of Quasilinear ordered set ;