[set theory] binary relation (example of binary relation on a | binary relation on a)

One 、 A Upper binary relation

---

A

A

A Upper binary relation :

yes

A

×

A

A \times A

A×A Any subset of the Cartesian product

R

R

R yes

A

A

A The binary relationship on

⇔

\Leftrightarrow

⇔

R

⊆

A

×

A

R \subseteq A \times A

R⊆A×A

⇔

\Leftrightarrow

⇔

R

∈

P

(

A

×

A

)

R \in P(A \times A)

R∈P(A×A)

Two 、 A The number of upper binary relations

---

aggregate

A

A

A The number of elements is

∣

A

∣

=

m

|A| = m

∣A∣=m

A

×

A

A \times A

A×A Cartesian product set in Ordered pair The number of elements is

∣

A

×

A

∣

=

m

2

|A \times A| = m^2

∣A×A∣=m2 individual ;

A

×

A

A \times A

A×A Cartesian product The number of power sets is

∣

P

(

A

×

A

)

∣

=

2

m

2

|P(A \times A)| = 2^{m^2}

∣P(A×A)∣=2m2

A

A

A The number of binary relations on has

2

m

2

2^{m^2}

2m2 individual ;

If

A

A

A There are

1

1

1 Elements ,

A

A

A The binary relation on has

2

1

2

=

2

2^{1^2} = 2

212=2 individual ;

If

A

A

A There are

2

2

2 Elements ,

A

A

A The binary relation on has

2

2

2

=

16

2^{2^2} = 16

222=16 individual ;

If

A

A

A There are

3

3

3 Elements ,

A

A

A The binary relation on has

2

3

2

=

512

2^{3^2} = 512

232=512 individual ;

3、 ... and 、 A Upper binary relation Example ( There are two elements in the set )

---

B

=

{

b

}

B = \{ b \}

B={ b}

aggregate

B

B

B The number of elements is

∣

B

∣

=

1

|B| = 1

∣B∣=1

B

×

B

B \times B

B×B Cartesian product set in Ordered pair The number of elements is

∣

B

×

B

∣

=

1

2

=

1

|B \times B| = 1^2 = 1

∣B×B∣=12=1 individual ;

B

×

B

B \times B

B×B Cartesian product The number of power sets is

∣

P

(

B

×

B

)

∣

=

2

1

2

=

2

|P(B \times B)| = 2^{1^2} = 2

∣P(B×B)∣=212=2

A

A

A The number of binary relations on has

2

1

2

=

2

2^{1^2} = 2

212=2 individual ;

0

0

0 individual Ordered pair The binary relationship of :

R

1

=

∅

R_1 = \varnothing

R1​=∅

1

1

1 individual Ordered pair The binary relationship of :

R

2

=

{

b

,

b

}

R_2 = \{ b , b \}

R2​={ b,b}

Four 、 A Upper binary relation Example ( There are two elements in the set )

---

aggregate

A

=

{

a

1

,

a

2

}

A = \{ a_1 , a_2 \}

A={ a1​,a2​}

be

A

A

A The binary relation on has

16

16

16 individual ;

A

×

A

A \times A

A×A Cartesian product set The number of ordered pairs in

4

4

4 individual ;

A

×

A

A \times A

A×A Cartesian product set The number of power sets is

2

4

=

16

2^4 = 16

24=16 ;

0

0

0 individual Ordered pair The binary relationship of :

1

1

1 individual

R

1

=

∅

R_1 = \varnothing

R1​=∅

1

1

1 individual Ordered pair The binary relationship of :

4

4

4 individual

R

2

=

{

a

1

,

a

1

}

R_2 = \{ a_1 , a_1 \}

R2​={ a1​,a1​}

R

3

=

{

a

1

,

a

2

}

R_3 = \{ a_1 , a_2 \}

R3​={ a1​,a2​}

R

4

=

{

a

2

,

a

1

}

R_4 = \{ a_2 , a_1 \}

R4​={ a2​,a1​}

R

5

=

{

a

2

,

a

2

}

R_5 = \{ a_2 , a_2 \}

R5​={ a2​,a2​}

2

2

2 individual Ordered pair The binary relationship of :

6

6

6 individual

R

6

=

{

{

a

1

,

a

1

}

,

{

a

1

,

a

2

}

}

R_6 = \{ \{ a_1 , a_1 \}, \{ a_1 , a_2 \} \}

R6​={ { a1​,a1​},{ a1​,a2​}}

R

7

=

{

{

a

1

,

a

1

}

,

{

a

2

,

a

1

}

}

R_7 = \{ \{ a_1 , a_1 \}, \{ a_2 , a_1 \} \}

R7​={ { a1​,a1​},{ a2​,a1​}}

R

8

=

{

{

a

1

,

a

1

}

,

{

a

2

,

a

2

}

}

R_8 = \{ \{ a_1 , a_1 \}, \{ a_2 , a_2 \} \}

R8​={ { a1​,a1​},{ a2​,a2​}}

R

9

=

{

{

a

1

,

a

2

}

,

{

a

2

,

a

1

}

}

R_9= \{ \{ a_1 , a_2 \} , \{ a_2 , a_1 \} \}

R9​={ { a1​,a2​},{ a2​,a1​}}

R

10

=

{

{

a

1

,

a

2

}

,

{

a

2

,

a

2

}

}

R_{10}= \{ \{ a_1 , a_2 \} , \{ a_2 , a_2 \} \}

R10​={ { a1​,a2​},{ a2​,a2​}}

R

11

=

{

{

a

2

,

a

1

}

,

{

a

2

,

a

2

}

}

R_{11}= \{ \{ a_2 , a_1 \} , \{ a_2 , a_2 \} \}

R11​={ { a2​,a1​},{ a2​,a2​}}

3

3

3 individual Ordered pair The binary relationship of :

4

4

4 individual

R

12

=

{

{

a

1

,

a

1

}

,

{

a

1

,

a

2

}

,

{

a

2

,

a

1

}

}

R_{12} = \{ \{ a_1 , a_1 \}, \{ a_1 , a_2 \} , \{ a_2 , a_1 \} \}

R12​={ { a1​,a1​},{ a1​,a2​},{ a2​,a1​}}

R

13

=

{

{

a

1

,

a

1

}

,

{

a

1

,

a

2

}

,

{

a

2

,

a

2

}

}

R_{13} = \{ \{ a_1 , a_1 \}, \{ a_1 , a_2 \} , \{ a_2 , a_2 \}\}

R13​={ { a1​,a1​},{ a1​,a2​},{ a2​,a2​}}

R

14

=

{

{

a

1

,

a

1

}

,

{

a

2

,

a

1

}

,

{

a

2

,

a

2

}

}

R_{14} = \{ \{ a_1 , a_1 \}, \{ a_2 , a_1 \} , \{ a_2 , a_2 \}\}

R14​={ { a1​,a1​},{ a2​,a1​},{ a2​,a2​}}

R

15

=

{

{

a

1

,

a

2

}

,

{

a

2

,

a

1

}

,

{

a

2

,

a

2

}

}

R_{15} = \{ \{ a_1 , a_2 \} , \{ a_2 , a_1 \} , \{ a_2 , a_2 \}\}

R15​={ { a1​,a2​},{ a2​,a1​},{ a2​,a2​}}

4

4

4 individual Ordered pair The binary relationship of :

1

1

1 individual

R

16

=

{

{

a

1

,

a

1

}

,

{

a

1

,

a

2

}

,

{

a

2

,

a

1

}

,

{

a

2

,

a

2

}

}

R_{16} = \{ \{ a_1 , a_1 \}, \{ a_1 , a_2 \} , \{ a_2 , a_1 \} , \{ a_2 , a_2 \}\}

R16​={ { a1​,a1​},{ a1​,a2​},{ a2​,a1​},{ a2​,a2​}}