当前位置：网站首页>[set theory] equivalence classes (concept of equivalence classes | examples of equivalence classes | properties of equivalence classes | quotient sets | examples of quotient sets)*
[set theory] equivalence classes (concept of equivalence classes | examples of equivalence classes | properties of equivalence classes | quotient sets | examples of quotient sets)*
2022-07-03 06:56:00 【Programmer community】
List of articles
- One 、 Equivalence class
- Two 、 Examples of equivalence classes
- 3、 ... and 、 Equivalence properties
- Four 、 Quotient set
- 5、 ... and 、 Quotient set example 1
- 6、 ... and 、 Quotient set example 2
- 7、 ... and 、 Quotient set example 3
One 、 Equivalence class
R
R
R Relationship yes
A
A
A aggregate The binary relationship on ,
A
A
A Set is not empty ,
A
≠
∅
A \not= \varnothing
A=∅ ,
about
A
A
A In the collection arbitrarily
x
x
x Elements ,
∀
x
∈
A
\forall x \in A
∀x∈A ,
x
x
x About
R
R
R Equivalence class of relation yes
[
x
]
R
=
{
y
∣
y
∈
A
∧
x
R
y
}
[x]_R = \{ y | y \in A \land xRy \}
[x]R={ y∣y∈A∧xRy} ;
x
x
x About
R
R
R Equivalence class of relation , Referred to as
x
x
x The equivalent class of , Write it down as
[
x
]
[x]
[x] ;
[
x
]
R
[x]_R
[x]R Express
x
x
x About
R
R
R Equivalence classes under relation ;
The equivalence class is composed of all And
x
x
x have
R
R
R Relational
y
y
y Set of components ;
If there is only one equivalence relation , Aforementioned
R
_R
R Subscripts can be omitted ,
[
x
]
R
[x]_R
[x]R It can be abbreviated as
[
x
]
[x]
[x]
Two 、 Examples of equivalence classes
aggregate
A
=
{
1
,
2
,
3
,
4
,
5
,
8
}
A = \{1,2,3,4,5,8\}
A={ 1,2,3,4,5,8}
R
R
R Relationship yes aggregate
A
A
A Upper model
3
3
3 Same as relation
Symbolized as :
R
=
<
x
,
y
>
∣
x
,
y
∈
A
∧
x
≡
y
(
m
o
d
3
)
R = {<x, y> | x, y \in A \land x \equiv y\pmod{3} }
R=<x,y>∣x,y∈A∧x≡y(mod3)
≡
\equiv
≡ The meaning of the symbol is Equal to
1
1
1 stay
R
R
R The equivalence class in relation is
{
1
,
4
}
\{ 1, 4 \}
{ 1,4}
2
2
2 stay
R
R
R The equivalence class in relation is
{
2
,
5
,
8
}
\{ 2, 5, 8 \}
{ 2,5,8}
3
3
3 stay
R
R
R The equivalence class in relation is
{
3
}
\{ 3 \}
{ 3}
Above
3
3
3 Equivalent classes , There is a global relationship inside the equivalence class , There is no relationship between equivalence classes ;
3、 ... and 、 Equivalence properties
R
R
R Relationship yes
A
A
A aggregate The equivalence relationship on ,
A
A
A Set is not empty ,
A
≠
∅
A \not= \varnothing
A=∅ , For any
A
A
A The elements in the collection
x
,
y
x,y
x,y ,
∀
x
,
y
∈
A
\forall x,y \in A
∀x,y∈A , It has the following properties :
① The equivalent class of each element is not empty ;
[
x
]
R
≠
∅
[x]_R \not= \varnothing
[x]R=∅
② If there is a relationship between the two elements , Then their equivalent classes are equal ;
x
R
y
⇒
[
x
]
R
=
[
y
]
R
xRy \Rightarrow [x]_R = [y]_R
xRy⇒[x]R=[y]R
③ If there is no relationship between the two elements , Then their equivalence classes must not intersect ;
¬
x
R
y
⇒
[
x
]
R
∩
[
y
]
R
=
∅
\lnot xRy \Rightarrow [x]_R \cap [y]_R = \varnothing
¬xRy⇒[x]R∩[y]R=∅
④ Union of all equivalence classes , Is the original set
A
A
A ;
⋃
{
[
x
]
R
∣
x
∈
A
}
=
A
\bigcup \{ [x]_R | x \in A \} = A
⋃{ [x]R∣x∈A}=A
Four 、 Quotient set
R
R
R Relationship yes
A
A
A aggregate The equivalence relationship on ,
A
A
A Set is not empty
A
A
A aggregate About
R
R
R Relationship The quotient set of yes
A
/
R
=
{
[
x
]
R
∣
x
∈
A
}
A/R = \{ [x]_R | x \in A \}
A/R={ [x]R∣x∈A}
abbreviation :
A
A
A The quotient set of
The essence of quotient set : Quotient set The essence is a aggregate , The elements in the set are Equivalence class , This equivalence class is based on
R
R
R Relational ;
5、 ... and 、 Quotient set example 1
aggregate
A
=
{
1
,
2
,
3
,
4
,
5
,
8
}
A = \{1,2,3,4,5,8\}
A={ 1,2,3,4,5,8}
R
R
R Relationship yes aggregate
A
A
A Upper model
3
3
3 Same as relation
Symbolized as :
R
=
<
x
,
y
>
∣
x
,
y
∈
A
∧
x
≡
y
(
m
o
d
3
)
R = {<x, y> | x, y \in A \land x \equiv y\pmod{3} }
R=<x,y>∣x,y∈A∧x≡y(mod3)
≡
\equiv
≡ The meaning of the symbol is Equal to
1
1
1 stay
R
R
R The equivalence class in relation is
{
1
,
4
}
\{ 1, 4 \}
{ 1,4}
2
2
2 stay
R
R
R The equivalence class in relation is
{
2
,
5
,
8
}
\{ 2, 5, 8 \}
{ 2,5,8}
3
3
3 stay
R
R
R The equivalence class in relation is
{
3
}
\{ 3 \}
{ 3}
Quotient set definition :
A
/
R
=
{
[
x
]
R
∣
x
∈
A
}
A/R = \{ [x]_R | x \in A \}
A/R={ [x]R∣x∈A}
A
A
A Set about
R
R
R The quotient set of relation is :
A
/
R
=
{
{
1
,
4
}
,
{
2
,
5
,
8
}
,
{
3
}
}
A/R = \{ \{ 1, 4 \} , \{ 2, 5, 8 \} , \{ 3 \} \}
A/R={ { 1,4},{ 2,5,8},{ 3}}
6、 ... and 、 Quotient set example 2
aggregate
A
=
{
a
1
,
a
2
,
⋯
,
a
n
}
A = \{ a_1 , a_2 , \cdots , a_n \}
A={ a1,a2,⋯,an} The equivalence relation on has :
I
A
I_A
IA Identity ,
E
A
E_A
EA Global relations ;
1. Identity
I
A
I_A
IA : Each element in the set is an equivalent class ; classification The smallest particle size ;
A
A
A Set about Identity
I
A
I_A
IA The quotient set of :
A
/
I
A
=
{
{
a
1
}
,
{
a
2
}
,
⋯
,
{
a
n
}
}
A/I_A = \{ \{ a_1 \} , \{ a_2 \} , \cdots , \{ a_n \} \}
A/IA={ { a1},{ a2},⋯,{ an}}
2. Global relations
E
A
E_A
EA : In the collection All elements are equivalent classes ; Put all the elements together , Every element has a relationship with each other ; This classification The coarsest particle size ;
A
A
A Set about Global relations
E
A
E_A
EA The quotient set of :
A
/
E
A
=
{
{
a
1
,
a
2
,
⋯
,
a
n
}
}
A/E_A = \{ \{ a_1 ,a_2 , \cdots , a_n \} \}
A/EA={ { a1,a2,⋯,an}}
3.
R
i
j
R_{ij}
Rij Relationship : Identity And
<
a
i
,
a
j
>
,
<
a
j
,
a
i
>
<a_i , a_j> , <a_j , a_i>
<ai,aj>,<aj,ai> Union ; The relationship is introspect , symmetry , Delivered , It's equivalence ;
R
i
j
R_{ij}
Rij Relationship description :
R
i
j
=
I
A
∪
{
<
a
i
,
a
j
>
,
<
a
j
,
a
i
>
}
R_{ij} = I_A \cup \{ <a_i , a_j> , <a_j , a_i> \}
Rij=IA∪{ <ai,aj>,<aj,ai>}
A
A
A Set about Global relations
R
i
j
R_{ij}
Rij The quotient set of :
- take
a
i
,
a
j
a_i, a_j
ai,aj In an equivalent class
{
a
i
,
a
j
}
\{ a_i , a_j \}
{
<
a
i
,
a
j
>
,
<
a
j
,
a
i
>
}
\{ <a_i , a_j> , <a_j , a_i> \}
{ <ai,aj>,<aj,ai>}
{ ai,aj}, Corresponding
- Put in the collection except
a
i
,
a
j
a_i, a_j
I
A
I_A
IA ,
{
a
1
}
,
⋯
,
{
a
i
−
1
}
,
{
a
i
+
1
}
,
⋯
,
{
a
j
−
1
}
,
{
a
j
+
1
}
,
⋯
,
a
n
}
\{a_1\} , \cdots , \{a_{i - 1}\}, \{a_{i + 1}\}, \cdots , \{a_{j - 1}\} , \{a_{j + 1}\}, \cdots , a_n \}
{ a1},⋯,{ ai−1},{ ai+1},⋯,{ aj−1},{ aj+1},⋯,an}
ai,aj Other elements besides are divided into a separate category , Corresponding
A
/
R
i
j
=
{
{
a
i
,
a
j
}
,
{
a
1
}
,
⋯
,
{
a
i
−
1
}
,
{
a
i
+
1
}
,
⋯
,
{
a
j
−
1
}
,
{
a
j
+
1
}
,
⋯
,
a
n
}
,
}
A/R_{ij} = \{ \{ a_i , a_j \} , \{a_1\} , \cdots , \{a_{i - 1}\}, \{a_{i + 1}\}, \cdots , \{a_{j - 1}\} , \{a_{j + 1}\}, \cdots , a_n \} , \}
A/Rij={ { ai,aj},{ a1},⋯,{ ai−1},{ ai+1},⋯,{ aj−1},{ aj+1},⋯,an},}
4. Empty relation
∅
\varnothing
∅ It's not a collection
A
A
A The equivalence relationship on , Empty relationships are not reflexive ;
7、 ... and 、 Quotient set example 3
aggregate
A
=
{
a
,
b
,
c
}
A = \{ a , b , c \}
A={ a,b,c} All equivalence relations on : share Five equivalence relations , Only Three elements , On the basis of identity , Consider two elements Between 2 In one direction Ordered pair composition The relationship between ;
①
R
1
=
I
A
R_1 = I_A
R1=IA Identity : The corresponding quotient set is :
A
/
I
A
=
{
{
a
}
,
{
b
}
,
{
c
}
}
A/I_A = \{ \{ a \} , \{ b \} , \{ c \} \}
A/IA={ { a},{ b},{ c}}
②
R
2
=
E
A
R_2 = E_A
R2=EA Global relations : The corresponding quotient set is :
A
/
E
A
=
{
{
a
,
b
,
c
}
}
A/E_A = \{ \{ a , b , c \} \}
A/EA={ { a,b,c}}
③
R
3
=
I
A
∪
{
<
b
,
c
>
,
<
c
,
b
>
}
R_3 = I_A \cup \{ <b,c>, <c,b> \}
R3=IA∪{ <b,c>,<c,b>} Relationship : The corresponding quotient set is :
A
/
R
3
=
{
{
a
}
,
{
b
,
c
}
}
A/R_3 = \{ \{ a \} , \{ b , c \} \}
A/R3={ { a},{ b,c}}
④
R
4
=
I
A
∪
{
<
a
,
c
>
,
<
c
,
a
>
}
R_4 = I_A \cup \{ <a,c>, <c,a> \}
R4=IA∪{ <a,c>,<c,a>} Relationship : The corresponding quotient set is :
A
/
R
4
=
{
{
b
}
,
{
a
,
c
}
}
A/R_4= \{ \{ b \} , \{ a , c \} \}
A/R4={ { b},{ a,c}}
⑤
R
5
=
I
A
∪
{
<
a
,
b
>
,
<
b
,
a
>
}
R_5 = I_A \cup \{ <a,b>, <b,a> \}
R5=IA∪{ <a,b>,<b,a>} Relationship : The corresponding quotient set is :
A
/
R
5
=
{
{
c
}
,
{
a
,
b
}
}
A/R_5 = \{ \{ c \} , \{ a , b \} \}
A/R5={ { c},{ a,b}}
边栏推荐
- Unit test notes
- 每日刷题记录 (十一)
- Numerical method for solving optimal control problem (I) -- gradient method
- Basic teaching of crawler code
- File links cannot be opened or downloaded in Google browser
- Opencv mouse and keyboard events
- Golang operation redis: write and read hash type data
- 【类和对象】深入浅出类和对象
- 熊市里的大机构压力倍增，灰度、Tether、微策略等巨鲸会不会成为'巨雷'？
- [classes and objects] explain classes and objects in simple terms
猜你喜欢
Liang Ning: 30 lectures on brain map notes for growth thinking
Ruoyi interface permission verification
Inno Setup 制作安装包
SQL implementation merges multiple rows of records into one row
Integration test practice (1) theoretical basis
【无标题】
2022 East China Normal University postgraduate entrance examination machine test questions - detailed solution
2022-06-23 VGMP-OSPF-域间安全策略-NAT策略(更新中)
Realize PDF to picture conversion with C ＃
The dynamic analysis and calculation of expressions are really delicious for flee
随机推荐
Golang operation redis: write and read hash type data
UTC time, GMT time, CST time
Redis command
MySQL installation
Dbnet: real time scene text detection with differentiable binarization
dataworks自定义函数开发环境搭建
What are the characteristics and functions of the scientific thinking mode of mechanical view and system view
mongodb
Know flex box
How can I split a string at the first occurrence of “-” (minus sign) into two $vars with PHP?
PHP install the spool extension
centos php7.3安装redis扩展
Application scenarios of Catalan number
Simple password lock
[open source project recommendation colugomum] this group of undergraduates open source retail industry solutions based on the domestic deep learning framework paddlepadddle
Shim and Polyfill in [concept collection]
如何迁移或复制VMware虚拟机系统
Ruoyi interface permission verification
2022年华东师范大学计科考研复试机试题-详细题解
Realize PDF to picture conversion with C ＃