[set theory] set concept and relationship (true subset | empty set | complete set | power set | number of set elements | power set steps)

One 、 True subset

---

True subset :

describe :

A

,

B

A , B

A,B Two sets , If

A

A

A aggregate yes

B

B

B A subset of a set , also

A

≠

B

A \not= B

A​=B , said

A

A

A yes

B

B

B The proper subset of ,

B

B

B It really includes

A

A

A ;

Write it down as :

A

⊂

B

A \subset B

A⊂B

Symbolize :

A

⊂

B

A \subset B

A⊂B

⇔

\Leftrightarrow

⇔

A

⊆

B

∧

A

≠

B

A \subseteq B \land A \not= B

A⊆B∧A​=B

Non proper subset :

describe :

A

A

A aggregate No

B

B

B The proper subset of a set ;

Write it down as :

A

⊄

B

A \not\subset B

A​⊂B

Symbolize :

A

⊄

B

A \not\subset B

A​⊂B

⇔

\Leftrightarrow

⇔

∃

x

(

x

∈

A

∧

x

∉

B

)

∧

A

≠

B

\exist x ( x \in A \land x \not\in B ) \land A \not= B

∃x(x∈A∧x​∈B)∧A​=B

( The element of being

x

x

x Is a collection

A

A

A The elements of , It's not a collection

B

B

B The elements of , also

A

,

B

A , B

A,B It's not equal , be

A

A

A No

B

B

B The proper subset of )

What a relationship nature :

Reflexivity :

A

⊄

A

A \not\subset A

A​⊂A

Antisymmetry : If

A

⊂

B

A \subset B

A⊂B , that

B

⊄

A

B \not\subset A

B​⊂A

Transitivity : If

A

⊂

B

A \subset B

A⊂B , also

B

⊂

C

B \subset C

B⊂C , that

A

⊂

C

A \subset C

A⊂C

Two 、 An empty set

---

Empty set description : A collection without any elements , It is called an empty set , Referred to as An empty set ;

Write it down as :

∅

\varnothing

∅

Empty set example :

A

=

{

x

∣

x

2

+

1

=

0

∧

x

∈

R

}

A = \{ x | x^2 + 1 = 0 \land x \in R \}

A={ x∣x2+1=0∧x∈R}

R

R

R Is a set of real numbers , Above

x

x

x Obviously, there is no solution , Set is also empty ;

Empty set theorem : An empty set is a subset of all sets ;

Empty set inference : Empty sets are unique ;

3、 ... and 、 The complete

---

The complete : Limit the set in question , Are subsets of a set , The set is called a complete set , Write it down as

E

E

E ;

It's not unique : The complete set is only relative to the scope of discussing problems , Is not the only , You cannot discuss situations outside the scope ;

Examples of complete works : Discuss [0, 1] Properties of real numbers on intervals , Take the complete set as [0, 1] All real numbers on ;

( Discuss the numbers of other intervals , You can also take other intervals as the complete set )

Four 、 Power set

---

Power set description :

A

A

A It's a collection ,

A

A

A A set consisting of all subsets of a set be called

A

A

A Power set of ;

Write it down as :

P

(

A

)

P(A)

P(A)

Symbolic expression :

P

(

A

)

=

{

x

∣

x

⊆

A

}

P(A) = \{ x | x \subseteq A \}

P(A)={ x∣x⊆A}

5、 ... and 、 Number of collection elements

---

Number of collection elements :

0

0

0 Meta set :

∅

\varnothing

∅

1

1

1 Meta set : contain

1

1

1 Collection of elements , Also known as Unit set ;

2

2

2 Meta set : contain

2

2

2 Collection of elements ;

⋮

\vdots

⋮

n

n

n Meta set : contain

n

n

n Collection of elements ; (

n

≥

1

n \geq 1

n≥1 )

There are poor sets :

∣

A

∣

|A|

∣A∣ Represents a collection

A

A

A The number of elements in , If

A

A

A The number of elements in the set is Finite number when , Then call it

A

A

A A set is a finite set , or Limited set ;

Theorem of the number of power sets : aggregate

A

A

A Medium Element number

∣

A

∣

=

n

|A| = n

∣A∣=n , be

A

A

A Of Number of power sets

∣

P

(

A

)

∣

=

2

n

|P(A)| = 2^n

∣P(A)∣=2n ;

6、 ... and 、 Power set steps

---

Power set steps : seek aggregate

A

A

A Power set of , It needs to be calculated in order

A

A

A Collection All subsets from low to high elements , Then these subsets are combined into a set ;

All subsets of low to high elements :

0

0

0 Meta set ,

1

1

1 Meta set ,

2

2

2 Meta set ,

⋯

\cdots

⋯ ,

n

n

n Meta set ;

aggregate

A

=

{

a

,

b

,

c

}

A = \{ a, b , c \}

A={ a,b,c}

0

0

0 Meta set :

∅

\varnothing

∅

1

1

1 Meta set :

{

a

}

\{ a \}

{ a} ,

{

b

}

\{ b \}

{ b} ,

{

c

}

\{ c \}

{ c}

2

2

2 Meta set :

{

a

,

b

}

\{ a, b \}

{ a,b} ,

{

a

,

c

}

\{ a, c \}

{ a,c} ,

{

b

,

c

}

\{ b, c \}

{ b,c}

3

3

3 Meta set :

{

a

,

b

,

c

}

\{ a, b, c \}

{ a,b,c}

aggregate

A

A

A The power set of is :

P

(

A

)

=

{

∅

,

{

a

}

,

{

b

}

,

{

c

}

,

{

a

,

b

}

,

{

a

,

c

}

,

{

b

,

c

}

,

{

a

,

b

,

c

}

}

P(A) = \{ \varnothing , \{ a \} , \{ b \} , \{ c \} , \{ a, b \} , \{ a, c \} , \{ b, c \} , \{ a, b, c \} \}

P(A)={ ∅,{ a},{ b},{ c},{ a,b},{ a,c},{ b,c},{ a,b,c}}