[set theory] binary relationship (special relationship type | empty relationship | identity relationship | global relationship | divisive relationship | 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 ;