当前位置:网站首页>[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
[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
2022-07-03 08:39:00 【Programmer community】
List of articles
- One 、 The whole sequence ( Line order relation )
- Two 、 Examples of total order relationships
- 3、 ... and 、 Coherent relations
- Four 、 Quasi ordered relation theorem 1
- Four 、 Quasi ordered relation theorem 2
- 5、 ... and 、 Trisomy 、 Quasilinear sequence
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
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
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 ;
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