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}}
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