[set theory] binary relationship (binary relationship notation | binary relationship from a to B | number of binary relationships | example of binary relationship)

One 、 Binary relationship

---

n

n

n Meta relationship :

Elements All are Orderly

n

n

n Set of tuples ;

n

n

n Meta relation example :

3 Meta relationship :

F

1

=

{

<

1

,

2

,

3

>

,

<

a

,

b

,

c

>

,

<

Count

learn

,

matter

The reason is

,

turn

learn

>

}

F_1 = \{ <1, 2, 3> , <a, b, c> , < mathematics , Physics , chemical > \}

F1​={ <1,2,3>,<a,b,c>,< Count learn , matter The reason is , turn learn >}

F

1

F_1

F1​ yes

3

3

3 Meta relationship , Each of its elements is Orderly

3

3

3 Tuples ;

4 Meta relationship :

F

2

=

{

<

1

,

2

,

3

,

4

>

,

<

a

,

b

,

c

,

d

>

,

<

language

writing

,

Count

learn

,

matter

The reason is

,

turn

learn

>

}

F_2 = \{ <1, 2, 3, 4> , <a, b, c, d> , < Chinese language and literature , mathematics , Physics , chemical > \}

F2​={ <1,2,3,4>,<a,b,c,d>,< language writing , Count learn , matter The reason is , turn learn >}

F

2

F_2

F2​ yes

4

4

4 Meta relationship , Each of its elements is Orderly

4

4

4 Tuples ;

Above order

n

n

n Tuples , The number is the same , The properties of elements can be different ;

Two 、 Binary relation notation

---

If

F

F

F It's a binary relationship (

F

F

F It's order

2

2

2 Tuple set )

Then there are :

<

x

,

y

>

∈

F

<x, y> \in F

<x,y>∈F

⇔

\Leftrightarrow

⇔

x

And

y

Yes

F

Turn off

system

x And y Yes F Relationship

x And y Yes F Turn off system

⇔

\Leftrightarrow

⇔

x

F

y

xFy

xFy

Binary relation notation :

① Infix notation ( infix ) :

x

F

y

xFy

xFy

② Prefix notation ( prefix ) :

F

(

x

,

y

)

F(x, y)

F(x,y) , or

F

x

y

Fxy

Fxy

③ Suffix notation ( suffix ) :

<

x

,

y

>

∈

F

<x,y> \in F

<x,y>∈F , or

x

y

F

xyF

xyF

Such as :

2

<

5

2 < 5

2<5 ,

2

2

2 Less than

5

5

5 ;

① Infix notation ( infix ) :

2

<

5

2 < 5

2<5

② Prefix notation ( prefix ) :

<

(

2

,

5

)

<(2, 5)

<(2,5)

③ Suffix notation ( suffix ) :

<

2

,

5

>

∈

<

<2,5> \in <

<2,5>∈<

3、 ... and 、 A To B The binary relationship of

---

A

A

A To

B

B

B The concept of binary relation :

A

×

B

A \times B

A×B Of Any subset yes

A

A

A To

B

B

B The binary relationship of

⇔

\Leftrightarrow

⇔

R

⊆

A

×

B

R \subseteq A \times B

R⊆A×B

⇔

\Leftrightarrow

⇔

R

∈

P

(

A

×

B

)

R \in P(A \times B)

R∈P(A×B)

A

A

A To

B

B

B The binary relationship of There may be

1

1

1 A collection of ,

2

2

2 A collection of ,

⋯

\cdots

⋯ ,

n

n

n A collection of ;

Four 、 A To B The number of binary relations

---

A

A

A To

B

B

B The number of binary relations :

∣

A

∣

=

m

|A| = m

∣A∣=m ,

∣

B

∣

=

n

|B| = n

∣B∣=n

A

A

A Number of collection elements

m

m

m individual ,

B

B

B Number of collection elements

n

n

n individual ;

Number of ordered pairs :

∣

A

×

B

∣

=

m

n

|A \times B| = mn

∣A×B∣=mn

Binary relationship Number :

∣

P

(

A

×

B

)

=

2

m

n

∣

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

∣P(A×B)=2mn∣, namely Above

m

n

mn

mn An ordered pair of total sets Power set Number ;

A

A

A To

B

B

B The number of binary relations =

A

×

B

A \times B

A×B Number of power sets =

2

m

n

2^{mn}

2mn individual

5、 ... and 、 A To B For example

---

A

=

{

a

1

,

a

2

}

A = \{a_1, a_2\}

A={ a1​,a2​} ,

B

=

{

b

}

B = \{ b \}

B={ b}

A

A

A aggregate And

B

B

B The Cartesian product of a set is :

A

×

B

=

{

∅

,

{

<

a

1

,

b

>

}

,

{

<

a

2

,

b

>

}

}

A \times B = \{ \varnothing, \{ <a_1 , b> \} , \{ <a_2 , b> \} \}

A×B={ ∅,{ <a1​,b>},{ <a2​,b>}}

analysis : Among them is

3

3

3 An orderly pair of , The number of binary relations is

2

2

×

1

=

4

2^{2 \times 1} = 4

22×1=4 individual , namely Above The power set of ordered pairs , Namely Yes

0

0

0 The number of ordered pairs

0

0

0 individual ,

1

1

1 The number of ordered pairs

2

2

2 individual ,

2

2

2 Number of ordered pairs

1

1

1 individual ;

A

A

A aggregate To

B

B

B A collection of Binary relationship : Yes

4

4

4 individual ;

R

1

=

∅

R_1 = \varnothing

R1​=∅ ,

a

1

a_1

a1​ And

b

b

b It doesn't matter. ,

a

2

a_2

a2​ And

b

b

b It doesn't matter. ;

R

2

=

{

<

a

1

,

b

>

}

R_2 = \{ <a_1 , b> \}

R2​={ <a1​,b>} ,

a

1

a_1

a1​ And

b

b

b It matters ,

a

2

a_2

a2​ And

b

b

b It doesn't matter. ;

R

3

=

{

<

a

2

,

b

>

}

R_3 = \{ <a_2 , b> \}

R3​={ <a2​,b>} ,

a

1

a_1

a1​ And

b

b

b It matters ,

a

2

a_2

a2​ And

b

b

b It doesn't matter. ;

R

4

=

{

<

a

1

,

b

>

,

<

a

2

,

b

>

}

R_4 = \{ <a_1 , b> , <a_2, b> \}

R4​={ <a1​,b>,<a2​,b>},

a

2

a_2

a2​ And

b

b

b It matters ,

a

1

a_1

a1​ And

b

b

b It matters ;

B

B

B aggregate And

A

A

A The Cartesian product of a set is :

A

×

B

=

{

∅

,

{

<

b

,

a

1

>

}

,

{

<

b

,

a

2

>

}

}

A \times B = \{ \varnothing, \{ <b, a_1 > \} , \{ <b, a_2 > \} \}

A×B={ ∅,{ <b,a1​>},{ <b,a2​>}}

analysis : Among them is

3

3

3 An orderly pair of , The number of binary relations is

2

2

×

1

=

4

2^{2 \times 1} = 4

22×1=4 individual , namely Above The power set of ordered pairs , Namely Yes

0

0

0 The number of ordered pairs

0

0

0 individual ,

1

1

1 The number of ordered pairs

2

2

2 individual ,

2

2

2 Number of ordered pairs

1

1

1 individual ;

B

B

B aggregate To

A

A

A A collection of Binary relationship : Yes

4

4

4 individual ;

R

5

=

∅

R_5 = \varnothing

R5​=∅ ,

b

b

b And

a

1

a_1

a1​ It doesn't matter. ,

b

b

b And

a

2

a_2

a2​ It doesn't matter. ;

R

6

=

{

<

b

,

a

1

>

}

R_6 = \{ <b, a_1 > \}

R6​={ <b,a1​>} ,

b

b

b And

a

1

a_1

a1​ It matters ,

b

b

b And

a

2

a_2

a2​ It doesn't matter. ;

R

7

=

{

<

b

,

a

2

>

}

R_7 = \{ <b, a_2> \}

R7​={ <b,a2​>} ,

b

b

b And

a

1

a_1

a1​ It doesn't matter. ,

b

b

b And

a

2

a_2

a2​ It matters ;

R

8

=

{

<

b

,

a

1

>

,

<

b

,

a

2

>

}

R_8 = \{ <b, a_1 > , <b, a_2> \}

R8​={ <b,a1​>,<b,a2​>} ,

b

b

b And

a

1

a_1

a1​ It matters ,

b

b

b And

a

2

a_2

a2​ It matters ;