[set theory] Cartesian product (concept of Cartesian product | examples of Cartesian product | properties of Cartesian product | non commutativity | non associativity | distribution law | ordered pair

Front blog : 【 Set theory 】 Ordered pair ( Ordered pair | Ordered triples | Orderly n Yuan Zu )

One 、 Cartesian product

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Cartesian product :

A

,

B

A , B

A,B It's two sets , from

A

A

A The element in the set is the first element , from

B

B

B The element in the set is the second element , A set of ordered pairs that meet the above conditions , Called a set

A

A

A And

B

B

B The Cartesian product of ;

Write it down as :

A

×

B

A \times B

A×B

Symbolize :

A

×

B

=

{

<

x

,

y

>

∣

x

∈

A

∧

y

∈

B

}

A \times B = \{ <x, y> | x \in A \land y \in B \}

A×B={ <x,y>∣x∈A∧y∈B}

aggregate

A

A

A And aggregate

B

B

B Of Cartesian product It's a New collection , This new set is a Ordered pair set ;

Two 、 Examples of Cartesian products

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aggregate

A

=

{

∅

,

a

}

A = \{ \varnothing , a \}

A={ ∅,a} , aggregate

B

=

{

1

,

2

,

3

}

B = \{ 1, 2, 3 \}

B={ 1,2,3}

A

×

B

=

{

<

∅

,

1

>

,

<

∅

,

2

>

,

<

∅

,

3

>

,

<

a

,

1

>

,

<

a

,

2

>

,

<

a

,

3

>

}

A \times B = \{ <\varnothing , 1> , <\varnothing , 2>, <\varnothing , 3>, <a, 1> , <a, 2> , <a , 3> \}

A×B={ <∅,1>,<∅,2>,<∅,3>,<a,1>,<a,2>,<a,3>}

Each ordered pair The first element comes from

A

A

A aggregate , The second element comes from

B

B

B aggregate ;

B

×

A

=

{

<

1

,

∅

>

,

<

2

,

∅

>

,

<

3

,

∅

>

,

<

1

,

a

>

,

<

2

,

a

>

,

<

3

,

a

>

}

B \times A = \{ <1, \varnothing > , <2, \varnothing >, <3 , \varnothing >, <1, a> , <2, a> , <3, a> \}

B×A={ <1,∅>,<2,∅>,<3,∅>,<1,a>,<2,a>,<3,a>}

The first element of each ordered pair comes from

B

B

B aggregate , The second element comes from

A

A

A aggregate ;

A

×

A

=

{

<

∅

,

∅

>

,

<

∅

,

a

>

,

<

a

,

∅

>

,

<

a

,

a

>

}

A \times A = \{< \varnothing, \varnothing> , <\varnothing, a> , <a, \varnothing> , <a, a> \}

A×A={ <∅,∅>,<∅,a>,<a,∅>,<a,a>}

The first element of each ordered pair comes from

A

A

A aggregate , The second element comes from

A

A

A aggregate ;

B

×

B

=

{

<

1

,

1

>

,

<

1

,

2

>

,

<

1

,

3

>

,

<

2

,

1

>

,

<

2

,

2

>

,

<

2

,

3

>

,

<

3

,

1

>

,

<

3

,

2

>

,

<

3

,

3

>

}

B \times B = \{ <1, 1> , <1, 2> , <1, 3> , <2, 1> , <2, 2> , <2,3> , <3,1> , <3,2> , <3,3> \}

B×B={ <1,1>,<1,2>,<1,3>,<2,1>,<2,2>,<2,3>,<3,1>,<3,2>,<3,3>}

The first element of each ordered pair comes from

B

B

B aggregate , The second element comes from

B

B

B aggregate ;

3、 ... and 、 Cartesian product property

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1. Non exchangeability

A

×

B

≠

B

×

A

A \times B \not= B \times A

A×B​=B×A

There are three special cases , Exchangeability is established

①

A

=

B

A = B

A=B

②

A

=

∅

A = \varnothing

A=∅

③

B

=

∅

B = \varnothing

B=∅

2. nonconjugal

(

A

×

B

)

×

C

≠

A

×

(

B

×

C

)

( A \times B ) \times C \not= A \times ( B \times C)

(A×B)×C​=A×(B×C)

There are three special cases , The combination is established

①

A

=

∅

A = \varnothing

A=∅

②

B

=

∅

B = \varnothing

B=∅

③

C

=

∅

C = \varnothing

C=∅

3. Distribution rate

A

×

(

B

∪

C

)

=

(

A

×

B

)

∪

(

A

×

C

)

A \times ( B \cup C ) = (A \times B) \cup (A \times C)

A×(B∪C)=(A×B)∪(A×C)

4. The case where the ordered pair is empty

A

×

B

=

∅

⇔

A

=

∅

∨

B

=

∅

A \times B = \varnothing \Leftrightarrow A = \varnothing \lor B= \varnothing

A×B=∅⇔A=∅∨B=∅

Four 、 n Vicat product

---

n Vicat product :

A

1

×

A

2

×

⋯

×

A

n

=

{

<

x

1

,

x

2

,

⋯
 

,

x

n

>

∣

x

1

∈

A

1

∧

x

2

∈

A

2

∧

⋯

∧

x

n

∈

A

n

}

A_1 \times A_2 \times \cdots \times A_n = \{ <x_1 , x_2, \cdots , x_n> | x_1 \in A_1 \land x_2 \in A_2 \land \cdots \land x_n \in A_n \}

A1​×A2​×⋯×An​={ <x1​,x2​,⋯,xn​>∣x1​∈A1​∧x2​∈A2​∧⋯∧xn​∈An​}

n

n

n The Cartesian product of sets ,

n

n

n Vicat product result , Each ordered pair has

n

n

n Elements , Each element is separate In the order specified From here

n

n

n A collection of ;

A

n

=

A

×

A

×

⋯

×

A

⏟

n

individual

A^n = \begin{matrix} \underbrace{ A \times A \times \cdots \times A } \\ n individual \end{matrix}

An=


A×A×⋯×A​n individual ​

This is a

n

n

n individual aggregate

A

A

A Of

n

n

n Vicat product ;

5、 ... and 、 n The number of Vicat's products

---

n

n

n The number of Vicat's products :

∣

A

i

∣

=

n

i

 

,

 

i

=

1

,

2

,

⋯
 

,

n

|A_i| = n_i \ , \ i = 1, 2, \cdots , n

∣Ai​∣=ni​ , i=1,2,⋯,n

⇒

\Rightarrow

⇒

∣

A

1

×

A

2

×

⋯

×

A

n

∣

=

n

1

×

n

2

×

⋯

×

n

n

| A_1 \times A_2 \times \cdots \times A_n | = n_1 \times n_2 \times \cdots \times n_n

∣A1​×A2​×⋯×An​∣=n1​×n2​×⋯×nn​

∣

A

i

∣

=

n

i

|A_i| = n_i

∣Ai​∣=ni​ ,

i

=

1

,

2

,

⋯
 

,

n

i = 1, 2, \cdots , n

i=1,2,⋯,n : Express The first

i

i

i A collection of

A

i

A_i

Ai​ The number of elements is

n

i

n_i

ni​ ;

∣

A

1

×

A

2

×

⋯

×

A

n

∣

| A_1 \times A_2 \times \cdots \times A_n |

∣A1​×A2​×⋯×An​∣ : Express

n

n

n The number of result sets of Cartesian products of sets ;

n

1

×

n

2

×

⋯

×

n

n

n_1 \times n_2 \times \cdots \times n_n

n1​×n2​×⋯×nn​ :

n

n

n The result of the Cartesian product of sets ;

6、 ... and 、 n Vicat product property

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n Vicat product property : And

2

2

2 Vicat's product has similar properties