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[set theory] binary relation (example of binary relation operation | example of inverse operation | example of composite operation | example of limiting operation | example of image operation)
2022-07-03 04:37:00 【Programmer community】
List of articles
- One 、 Examples of inverse operations
- Two 、 Examples of composite operations ( Reverse order synthesis )
- 3、 ... and 、 Examples of limiting operations
- Four 、 Like operation examples
One 、 Examples of inverse operations
A
=
{
a
,
b
,
c
,
d
}
A = \{ a, b, c, d \}
A={ a,b,c,d}
B
=
{
a
,
b
,
<
c
,
d
>
}
B = \{ a, b, <c, d> \}
B={ a,b,<c,d>}
C
=
{
<
a
,
b
>
,
<
c
,
d
>
}
C = \{ <a, b> , <c, d> \}
C={ <a,b>,<c,d>}
Find the inverse operation of the above set
The inverse operation can only aim at Ordered pair Conduct , If there is no order, right , There is no concept of relational operation ;
A
A
A There are no ordered pairs in the set , Therefore, there is no concept of relational operation , Inverse it , The result is an empty set ;
A
−
1
=
∅
A^{-1} = \varnothing
A−1=∅
B
B
B Collection Yes Ordered pair
<
c
,
d
>
<c, d>
<c,d> , The inverse operation is to find the inverse of all ordered pairs ;
B
−
1
=
{
<
d
,
c
>
}
B^{-1} = \{ <d, c> \}
B−1={ <d,c>}
C
C
C Collection Yes Ordered pair
<
a
,
b
>
,
<
c
,
d
>
<a,b> , <c, d>
<a,b>,<c,d> , The inverse operation is to find the inverse of all ordered pairs ;
C
−
1
=
{
<
b
,
a
>
,
<
d
,
c
>
}
C^{-1} = \{ <b,a> , <d, c> \}
C−1={ <b,a>,<d,c>}
Two 、 Examples of composite operations ( Reverse order synthesis )
B
=
{
a
,
b
,
<
c
,
d
>
}
B = \{ a, b , <c,d> \}
B={ a,b,<c,d>}
R
=
{
<
a
,
b
>
,
<
c
,
d
>
}
R = \{ <a,b> , <c,d> \}
R={ <a,b>,<c,d>}
G
=
{
<
b
,
e
>
,
<
d
,
c
>
}
G = \{ <b, e> , <d, c> \}
G={ <b,e>,<d,c>}
Find the result of the following synthesis operation , there synthesis refer to Reverse order synthesis
B
o
R
−
1
B o R^{-1}
BoR−1
R
−
1
=
{
<
b
,
a
>
,
<
d
,
c
>
}
R^{-1} = \{ <b,a> , <d,c> \}
R−1={ <b,a>,<d,c>}
B
o
R
−
1
=
{
<
c
,
d
>
}
o
{
<
b
,
a
>
,
<
d
,
c
>
}
=
{
<
d
,
d
>
}
B o R^{-1} = \{ <c, d> \} o \{ <b,a> , <d,c> \} = \{ <d, d> \}
BoR−1={ <c,d>}o{ <b,a>,<d,c>}={ <d,d>}
synthesis The default is Reverse order synthesis
G
o
B
G o B
GoB
G
o
B
=
{
<
b
,
e
>
,
<
d
,
c
>
}
o
{
<
c
,
d
>
}
=
{
<
c
,
c
>
}
G o B = \{<b,e>, <d, c>\} o \{ <c,d> \} = \{ <c,c> \}
GoB={ <b,e>,<d,c>}o{ <c,d>}={ <c,c>}
G
o
R
G o R
GoR
G
o
R
=
{
<
b
,
e
>
,
<
d
,
c
>
}
o
{
<
a
,
b
>
,
<
c
,
d
>
}
=
{
<
a
,
e
>
,
<
c
,
c
>
}
G o R =\{<b,e>, <d, c>\} o \{ <a,b> , <c,d> \} = \{ <a,e>, <c,c> \}
GoR={ <b,e>,<d,c>}o{ <a,b>,<c,d>}={ <a,e>,<c,c>}
R
o
G
R o G
RoG
R
o
G
=
{
<
a
,
b
>
,
<
c
,
d
>
}
o
{
<
b
,
e
>
,
<
d
,
c
>
}
=
{
<
d
,
d
>
}
R o G =\{ <a,b> , <c,d> \} o \{<b,e>, <d, c>\} = \{ <d,d> \}
RoG={ <a,b>,<c,d>}o{ <b,e>,<d,c>}={ <d,d>}
3、 ... and 、 Examples of limiting operations
F
=
{
<
a
,
b
>
,
<
a
,
{
a
}
>
,
<
{
a
}
,
{
a
,
{
a
}
}
>
}
F = \{ <a,b> , <a, \{a\}> , <\{a\} , \{a, \{a\}\}> \}
F={ <a,b>,<a,{ a}>,<{ a},{ a,{ a}}>}
Reference resources : 【 Set theory 】 Binary relationship ( Domain of definition | range | Domain | Inverse operation | Reverse composition operation | Limit | image | Single root | Single value | The nature of composition operation ) 5、 ... and 、 Relationship constraints
1. seek
F
↾
{
a
}
F \upharpoonright \{a\}
F↾{ a}
F
F
F Ordered pairs in sets , The first element is
{
a
}
\{a\}
{ a} An ordered pair of elements in a set , The set of these ordered pairs is
F
F
F aggregate stay
{
a
}
\{a\}
{ a} Restrictions on sets ;
F
↾
{
a
}
=
{
<
a
,
b
>
,
<
a
,
{
a
}
>
}
F \upharpoonright \{a\} = \{ <a,b> , <a, \{a\}> \}
F↾{ a}={ <a,b>,<a,{ a}>}
2. seek
F
↾
{
{
a
}
}
F \upharpoonright \{\{a\}\}
F↾{ { a}}
F
F
F Ordered pairs in sets , The first element is
{
{
a
}
}
\{\{a\}\}
{ { a}} An ordered pair of elements in a set ,
{
{
a
}
}
\{\{a\}\}
{ { a}} The elements in the set are
{
a
}
\{a\}
{ a} , The set of these ordered pairs is
F
F
F aggregate stay
{
{
a
}
}
\{\{a\}\}
{ { a}} Restrictions on sets ;
F
↾
{
{
a
}
}
=
{
<
{
a
,
{
a
}
}
>
}
F \upharpoonright \{\{a\}\} = \{ <\{a, \{a\}\}> \}
F↾{ { a}}={ <{ a,{ a}}>}
3. seek
F
↾
{
a
,
{
a
}
}
F \upharpoonright \{a, \{a\}\}
F↾{ a,{ a}}
F
F
F Ordered pairs in sets , The first element is
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} The elements in the collection That's right , The set of these ordered pairs is
F
F
F aggregate stay
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} Restrictions on sets ;
F
↾
{
a
,
{
a
}
}
=
{
<
a
,
b
>
,
<
a
,
{
a
}
>
,
<
{
a
}
,
{
a
,
{
a
}
}
>
}
F \upharpoonright \{a, \{a\}\} = \{ <a,b> , <a, \{a\}> , <\{a\} , \{a, \{a\}\}> \}
F↾{ a,{ a}}={ <a,b>,<a,{ a}>,<{ a},{ a,{ a}}>}
4. seek
F
−
1
↾
{
{
a
}
}
F^{-1} \upharpoonright \{\{a\}\}
F−1↾{ { a}}
F
−
1
=
{
<
b
,
a
>
,
<
{
a
}
,
a
>
,
<
{
a
,
{
a
}
}
,
{
a
}
>
}
F^{-1} = \{ <b, a> , <\{a\}, a> , <\{a, \{a\}\}, \{a\} > \}
F−1={ <b,a>,<{ a},a>,<{ a,{ a}},{ a}>}
F
−
1
F^{-1}
F−1 Ordered pairs in sets , The first element is
{
{
a
}
}
\{\{a\}\}
{ { a}} The elements in the collection That's right , The set of these ordered pairs is
F
−
1
F^{-1}
F−1 aggregate stay
{
{
a
}
}
\{\{a\}\}
{ { a}} Restrictions on sets ;
F
−
1
↾
{
{
a
}
}
=
{
<
{
a
}
,
a
>
}
F^{-1} \upharpoonright \{\{a\}\} = \{ <\{a\}, a> \}
F−1↾{ { a}}={ <{ a},a>}
Four 、 Like operation examples
F
=
{
<
a
,
b
>
,
<
a
,
{
a
}
>
,
<
{
a
}
,
{
a
,
{
a
}
}
>
}
F = \{ <a, b> , <a, \{ a \}> , <\{ a \} , \{ a, \{a\} \}> \}
F={ <a,b>,<a,{ a}>,<{ a},{ a,{ a}}>}
Reference resources : 【 Set theory 】 Binary relationship ( Domain of definition | range | Domain | Inverse operation | Reverse composition operation | Limit | image | Single root | Single value | The nature of composition operation ) 6、 ... and 、 Image of relationship
F
F
F Assemble in
A
A
A Collective image , yes
F
F
F Assemble in
A
A
A Limited on the set range ;
1.
F
[
{
a
}
]
F[\{a\}]
F[{ a}]
F
F
F Assemble in
{
a
}
\{a\}
{ a} The image on the set , yes
F
F
F Assemble in
{
a
}
\{a\}
{ a} The value range of the restriction on the set ,
F
F
F Assemble in
{
a
}
\{a\}
{ a} The limit on the set is
{
<
a
,
b
>
,
<
a
,
{
a
}
>
}
\{ <a, b> , <a, \{ a \}> \}
{ <a,b>,<a,{ a}>} , Corresponding
F
F
F Assemble in
{
a
}
\{a\}
{ a} On the set, it looks like
{
b
,
{
a
}
}
\{ b, \{a\} \}
{ b,{ a}}
F
[
{
a
}
]
=
{
b
,
{
a
}
}
F[\{a\}] = \{ b, \{a\} \}
F[{ a}]={ b,{ a}}
2.
F
[
{
a
,
{
a
}
}
]
F[\{a, \{a\}\}]
F[{ a,{ a}}]
F
F
F Assemble in
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} The image on the set , yes
F
F
F Assemble in
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} The value range of the restriction on the set ,
F
F
F Assemble in
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} The limit on the set is
{
<
a
,
b
>
,
<
a
,
{
a
}
>
,
<
{
a
}
,
{
a
,
{
a
}
}
>
}
\{ <a, b> , <a, \{ a \}> , <\{ a \} , \{ a, \{a\} \}> \}
{ <a,b>,<a,{ a}>,<{ a},{ a,{ a}}>} , Corresponding
F
F
F Assemble in
{
a
,
{
a
}
}
\{a, \{a\}\}
{ a,{ a}} On the set, it looks like
{
b
,
{
a
}
,
{
a
,
{
a
}
}
\{ b, \{a\} , \{ a, \{a\} \}
{ b,{ a},{ a,{ a}}
F
[
{
a
,
{
a
}
}
]
=
{
b
,
{
a
}
,
{
a
,
{
a
}
}
F[\{a, \{a\}\}] = \{ b, \{a\} , \{ a, \{a\} \}
F[{ a,{ a}}]={ b,{ a},{ a,{ a}}
3.
F
−
1
[
{
a
}
]
F^{-1}[\{a\}]
F−1[{ a}]
F
−
1
=
{
<
b
,
a
>
,
<
{
a
}
,
a
>
,
<
{
a
,
{
a
}
}
,
{
a
}
>
}
F^{-1} = \{ <b, a> , <\{a\}, a> , <\{a, \{a\}\}, \{a\} > \}
F−1={ <b,a>,<{ a},a>,<{ a,{ a}},{ a}>}
F
−
1
F^{-1}
F−1 Assemble in
{
a
}
\{a\}
{ a} The image on the set , yes
F
−
1
F^{-1}
F−1 Assemble in
{
a
}
\{a\}
{ a} The value range of the restriction on the set ,
F
−
1
F^{-1}
F−1 Assemble in
{
a
}
\{a\}
{ a} The limit on the set is
∅
\varnothing
∅ , Corresponding
F
−
1
F^{-1}
F−1 Assemble in
{
a
}
\{a\}
{ a} On the set, it looks like
∅
\varnothing
∅
F
−
1
[
{
a
}
]
=
∅
F^{-1}[\{a\}] = \varnothing
F−1[{ a}]=∅
4.
F
−
1
[
{
{
a
}
}
]
F^{-1}[\{ \{a\} \}]
F−1[{ { a}}]
F
−
1
=
{
<
b
,
a
>
,
<
{
a
}
,
a
>
,
<
{
a
,
{
a
}
}
,
{
a
}
>
}
F^{-1} = \{ <b, a> , <\{a\}, a> , <\{a, \{a\}\}, \{a\} > \}
F−1={ <b,a>,<{ a},a>,<{ a,{ a}},{ a}>}
F
−
1
F^{-1}
F−1 Assemble in
{
{
a
}
}
\{ \{a\} \}
{ { a}} The image on the set , yes
F
−
1
F^{-1}
F−1 Assemble in
{
{
a
}
}
\{ \{a\} \}
{ { a}} The value range of the restriction on the set ,
F
−
1
F^{-1}
F−1 Assemble in
{
{
a
}
}
\{ \{a\} \}
{ { a}} The limit on the set is
<
{
a
}
,
a
>
<\{a\}, a>
<{ a},a> , Corresponding
F
−
1
F^{-1}
F−1 Assemble in
{
{
a
}
}
\{ \{a\} \}
{ { a}} On the set, it looks like
{
a
}
\{a\}
{ a}
F
−
1
[
{
{
a
}
}
]
=
{
a
}
F^{-1}[\{ \{a\} \}] = \{a\}
F−1[{ { a}}]={ a}
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