当前位置:网站首页>[set theory] binary relationship (special relationship type | empty relationship | identity relationship | global relationship | divisive relationship | size relationship)
[set theory] binary relationship (special relationship type | empty relationship | identity relationship | global relationship | divisive relationship | size relationship)
2022-07-03 04:28:00 【Programmer community】
List of articles
- One 、 A special relationship
- Two 、 Special relationships on sets
- 3、 ... and 、 Division relations
- Four 、 Size relationship
One 、 A special relationship
Special binary relationship :
- Empty relation
- Identity
- Global relations
- Division relations
- Less than or equal to
- Inclusion relation
- What a relationship
Two 、 Special relationships on sets
aggregate
A
A
A Is an arbitrary set , aggregate
A
A
A The following relationships can be defined in :
Empty relation :
∅
\varnothing
∅ , There is no relation in empty relation ;
Identity :
I
A
=
{
<
x
,
x
>
∣
x
∈
A
}
I_A = \{ <x, x> | x \in A \}
IA={ <x,x>∣x∈A}
Global relations :
E
A
=
A
×
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
}
E_A = A \times A = \{ <x,y> | x \in A \land y \in A \}
EA=A×A={ <x,y>∣x∈A∧y∈A} , There is a relationship between any two elements ;
The above three relationships are the most basic relationships , Any set can define the above three relationships ;
Global relations yes The biggest relationship , It contains all possible ordered pairs ;
Empty relation yes Minimal relationship , There is no orderly pair ;
Identity It has a special meaning , It doesn't play any role in relational operation ;
3、 ... and 、 Division relations
A
⊆
Z
A \subseteq Z
A⊆Z ,
A
A
A A set is a subset of an integer set , Definition
A
A
A Divisible relation on set :
D
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
∧
x
∣
y
}
D_A = \{ <x, y> | x \in A \land y \in A \land x|y \}
DA={ <x,y>∣x∈A∧y∈A∧x∣y}
Among them
x
∣
y
x|y
x∣y Medium
∣
|
∣ The symbol means division ,
x
x
x to be divisible by
y
y
y ;
x
x
x to be divisible by
y
y
y ,
x
x
x It's a divisor ( The denominator ) ,
y
y
y It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
xy
y
y
y Can be
x
x
x to be divisible by ,
x
x
x It's a divisor ( The denominator ) ,
y
y
y It's a dividend ( molecular ) ;
y
x
\dfrac{y}{x}
xy
Examples of divisible relationships :
A
=
{
1
,
2
,
3
,
4
}
A = \{ 1, 2, 3, 4 \}
A={ 1,2,3,4}
D
A
=
{
<
1
,
1
>
,
<
1
,
2
>
,
<
1
,
3
>
,
<
1
,
4
>
,
<
2
,
2
>
,
<
2
,
4
>
,
<
3
,
3
>
,
<
4
,
4
>
}
D_A = \{ <1, 1> , <1, 2> , <1, 3> , <1, 4> , <2, 2> , <2, 4> , <3, 3> , <4,4> \}
DA={ <1,1>,<1,2>,<1,3>,<1,4>,<2,2>,<2,4>,<3,3>,<4,4>}
Four 、 Size relationship
A
⊆
R
A \subseteq R
A⊆R , aggregate
A
A
A Is a subset of the set of real numbers , In collection
A
A
A There is the following binary relationship :
Greater than the relationship ( Great Than ) :
G
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
∧
x
>
y
}
G_A = \{ <x,y> | x \in A \land y \in A \land x > y \}
GA={ <x,y>∣x∈A∧y∈A∧x>y}
Relationship greater than or equal to ( Great Than Or Equal To ) :
G
E
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
∧
x
≥
y
}
GE_A = \{ <x,y> | x \in A \land y \in A \land x \geq y \}
GEA={ <x,y>∣x∈A∧y∈A∧x≥y}
Less than relationship ( Less Than ) :
L
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
∧
x
<
y
}
L_A = \{ <x,y> | x \in A \land y \in A \land x < y \}
LA={ <x,y>∣x∈A∧y∈A∧x<y}
Less than or equal to ( Less Than Or Equal To ) :
L
E
A
=
{
<
x
,
y
>
∣
x
∈
A
∧
y
∈
A
∧
x
≤
y
}
LE_A = \{ <x,y> | x \in A \land y \in A \land x \leq y \}
LEA={ <x,y>∣x∈A∧y∈A∧x≤y}
If
A
A
A Set is a finite set , be
A
A
A The relationship on is limited ;
If
A
A
A A set is an infinite set , be
A
A
A The relationship on is infinite ;
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