当前位置：网站首页>[set theory] set operation (Union | intersection | disjoint | relative complement | symmetric difference | absolute complement | generalized union | generalized intersection | set operation priority)

[set theory] set operation (Union | intersection | disjoint | relative complement | symmetric difference | absolute complement | generalized union | generalized intersection | set operation priority)

2022-07-03 04:09:00 【Programmer community】

List of articles

One 、 Combine
Two 、 Union example
3、 ... and 、 intersection
Four 、 Example of intersection
5、 ... and 、 Disjoint
6、 ... and 、 Relative complement
7、 ... and 、 Symmetry difference
8、 ... and 、 Absolute complement
Nine 、 Generalized Union
Ten 、 Generalized intersection
11、 ... and 、 Set operation priority

One 、 Combine

Combine :

A, B

$A, B$ It's two sets , from

$A$ and

$B$ A collection of all elements , be called

$A$ And

$B$ Union ;

Remember to do :

∪

A \cup B

$A \cup B$ ,

∪

\cup

$\cup$ be called Union operator ;

Symbolize :

∪

{

∣

∈

∨

∈

}

A \cup B = \{ x | x \in A \lor x \in B \}

$A \cup B = {x ∣ x \in A \lor x \in B}$

Primary union : The union of two sets , It can be extended to A limited number / Countable The union of sets , be called Primary union ;

⋯

A_1 , A_2 , \cdots , A_n

$A_{1}, A_{2}, \dots, A_{n}$ yes

$n$ A collection of , be

∪

⋯

∪

{

∣

∃

(

≤

∨

∈

)

}

A_1 \cup A_2 \cup \cdots \cup A_n = \{ x | \exist i ( 1 \leq i \leq n \ \lor \ x \in A_i ) \}

$A_{1} \cup A_{2} \cup \dots \cup A_{n} = {x ∣ \exists i (1 \leq i \leq n \lor x \in A_{i})}$ , Write it down as

⋃

∪

⋯

∪

\bigcup_{i=1}^{n} A_i = A_1 \cup A_2 \cup \cdots \cup A_n

$i = 1 ⋃ n A_{i} = A_{1} \cup A_{2} \cup \dots \cup A_{n}$

⋯

A_1 , A_2 , \cdots , A_n , \cdots

$A_{1}, A_{2}, \dots, A_{n}, \dots$ yes Countable A collection of , be The primary union form is recorded as :

⋃

∞

∪

⋯

\bigcup_{i=1}^{\infty} A_i = A_1 \cup A_2 \cup \cdots

$i = 1 ⋃ \infty A_{i} = A_{1} \cup A_{2} \cup \dots$

Two 、 Union example

aggregate

{

∈

∣

≤

}

A = \{ x \in N | 5 \leq x \leq 10 \}

$A = {x \in N ∣ 5 \leq x \leq 10}$ , aggregate

{

∈

∣

≤

∨

yes

plain

Count

}

B = \{ x \in N | x \leq 10 \lor x Prime number \}

$B = {x \in N ∣ x \leq 10 \lor x yes plain Count}$

∪

{

}

A \cup B = \{ 2, 3, 5 ,6,7,8,9,10 \}

$A \cup B = {2, 3, 5, 6, 7, 8, 9, 10}$

3、 ... and 、 intersection

intersection :

A, B

$A, B$ It's two sets ,

$A$ and

$B$ A collection of common elements , be called

A , B

$A, B$ Intersection of sets ;

Write it down as :

∩

A \cap B

$A \cap B$ ,

∩

\cap

$\cap$ be called Transportation operator ;

Symbolize :

∩

{

∣

∈

∧

∈

}

A \cap B = \{ x | x \in A \land x \in B \}

$A \cap B = {x ∣ x \in A \land x \in B}$

Primary delivery : Intersection of two sets , It can be extended to A limited number / Countable The union of sets , be called Primary delivery ;

⋯

A_1 , A_2 , \cdots , A_n

$A_{1}, A_{2}, \dots, A_{n}$ yes

$n$ A collection of , be

∩

⋯

∩

{

∣

∀

(

≤

→

∈

)

}

A_1 \cap A_2 \cap \cdots \cap A_n = \{ x | \forall i ( 1 \leq i \leq n \ \to \ x \in A_i ) \}

$A_{1} \cap A_{2} \cap \dots \cap A_{n} = {x ∣ \forall i (1 \leq i \leq n \to x \in A_{i})}$ , Write it down as

⋂

∩

⋯

∩

\bigcap_{i=1}^{n} A_i = A_1 \cap A_2 \cap \cdots \cap A_n

$i = 1 ⋂ n A_{i} = A_{1} \cap A_{2} \cap \dots \cap A_{n}$

⋯

A_1 , A_2 , \cdots , A_n , \cdots

$A_{1}, A_{2}, \dots, A_{n}, \dots$ yes Countable A collection of , be The primary union form is recorded as :

⋂

∞

∩

⋯

\bigcap_{i=1}^{\infty} A_i = A_1 \cap A_2 \cap \cdots

$i = 1 ⋂ \infty A_{i} = A_{1} \cap A_{2} \cap \dots$

Four 、 Example of intersection

aggregate

{

∈

∣

≤

}

A = \{ x \in N | 5 \leq x \leq 10 \}

$A = {x \in N ∣ 5 \leq x \leq 10}$ , aggregate

{

∈

∣

≤

∧

yes

plain

Count

}

B = \{ x \in N | x \leq 10 \land x Prime number \}

$B = {x \in N ∣ x \leq 10 \land x yes plain Count}$

∩

{

}

A \cap B = \{ 5, 7 \}

$A \cap B = {5, 7}$

5、 ... and 、 Disjoint

Disjoint :

A , B

$A, B$ Two sets , If

∩

∅

A \cap B = \varnothing

$A \cap B = \emptyset$ , said

$A$ and

$B$ The two sets are Disjoint Of ;

Extend to multiple collections :

⋯

A_1 , A_2 , \cdots

$A_{1}, A_{2}, \dots$ Is a countable set , arbitrarily

≠

i \not= j

$i \neq = j$ ,

∩

∅

A_i \cap A_j = \varnothing

$A_{i} \cap A_{j} = \emptyset$ All set up , said

⋯

A_1 , A_2 , \cdots

$A_{1}, A_{2}, \dots$ Are disjoint ;

6、 ... and 、 Relative complement

Relative complement :

A , B

$A, B$ Two sets , Belong to

$A$ aggregate and Do not belong to

$B$ aggregate Of A collection of all elements , be called

$B$ Yes

$A$ The relative complement of ;

Write it down as :

−

A - B

$A - B$

Symbolize :

−

{

∣

∈

∧

∉

}

A-B = \{ x | x \in A \land x \not\in B \}

$A - B = {x ∣ x \in A \land x \neq \in B}$

7、 ... and 、 Symmetry difference

Symmetry difference :

A , B

$A, B$ It's two sets , Belong to

$A$ aggregate and Do not belong to

$B$ aggregate , Belong to

$B$ aggregate and Do not belong to

$A$ aggregate , Of All elements , The set formed is called

$A$ And

$B$ Symmetry difference of ;

Write it down as :

⊕

A \oplus B

$A \oplus B$

Symbolize :

⊕

{

∣

(

∈

∧

∉

)

∨

(

∉

∧

∈

)

}

A \oplus B = \{ x | ( x \in A \land x \not\in B ) \lor ( x \not\in A \land x \in B ) \}

$A \oplus B = {x ∣ (x \in A \land x \neq \in B) \lor (x \neq \in A \land x \in B)}$

Symmetry difference And Relative complement Relationship :

⊕

(

−

)

∪

(

−

)

(

∪

)

−

(

∩

)

A \oplus B = ( A - B ) \cup ( B - A ) = ( A \cup B ) - ( A \cap B )

$A \oplus B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$

(

−

)

∪

(

−

)

( A - B ) \cup ( B - A )

$(A - B) \cup (B - A)$ :

$A$ Yes

$B$ The relative complement of , And

$B$ Yes

$A$ The relative complement of Of Combine ;

(

∪

)

−

(

∩

)

( A \cup B ) - ( A \cap B )

$(A \cup B) - (A \cap B)$ :

A, B

$A, B$ Union Yes

A,B

$A, B$ Relative complement of intersection ;

8、 ... and 、 Absolute complement

Absolute complement :

$E$ It's the whole book ,

⊆

A \subseteq E

$A \subseteq E$ , The complete

$E$ contain

$A$ aggregate , call

$A$ Yes

$E$ The relative complement of by

$A$ The absolute complement of ;

Write it down as :

∼

\sim A

$\sim A$

Symbolize :

∼

{

∣

∈

∧

∉

}

\sim A = \{ x | x \in E \land x \not\in A \}

$\sim A = {x ∣ x \in E \land x \neq \in A}$

among

$E$ It's the whole book ,

∈

x \in E

$x \in E$ To be eternal , according to Propositional logic Equivalent calculus Of The same thing ,

$1$ Syntaxis Any value , The true value is still Any value In itself ;

therefore , Sure Get rid of Conjunctions Ahead

∈

x \in E

$x \in E$ , The result is :

∼

{

∣

∉

}

\sim A = \{ x | x \not\in A \}

$\sim A = {x ∣ x \neq \in A}$

Nine 、 Generalized Union

Generalized Union :

\mathscr{A}

$A$ It's a Set family , Set family

\mathscr{A}

$A$ All in Collection elements Of A collection of elements , be called Set family

\mathscr{A}

$A$ Generalized union of ;

Write it down as :

∪

\cup \mathscr{A}

$\cup A$

Symbolize :

∪

{

∣

∃

(

∈

∧

∈

)

}

\cup \mathscr{A} = \{ x | \exist z ( x \in z \land z \in \mathscr{A} ) \}

$\cup A = {x ∣ \exists z (x \in z \land z \in A)}$

Examples of generalized union :

{

}

{

}

{

}

\mathscr{A} = \{ \{a, b\} , \{a, c\} , \{a, b, c\} \}

$A = {{a, b}, {a, c}, {a, b, c}}$

∪

{

}

\cup \mathscr{A} = \{ a, b, c \}

$\cup A = {a, b, c}$

Ten 、 Generalized intersection

Generalized intersection :

\mathscr{A}

$A$ It's a Set family , Set family

\mathscr{A}

$A$ All in Collection elements Of A collection of common elements , be called Set family

\mathscr{A}

$A$ Generalized intersection of ;

Write it down as :

∩

\cap \mathscr{A}

$\cap A$

Symbolize :

∩

{

∣

∀

(

∈

→

∈

)

}

\cap \mathscr{A} = \{ x | \forall z ( z \in \mathscr{A} \to x \in z ) \}

$\cap A = {x ∣ \forall z (z \in A \to x \in z)}$

Examples of generalized union :

{

}

{

}

{

}

\mathscr{A} = \{ \{a, b\} , \{a, c\} , \{a, b, c\} \}

$A = {{a, b}, {a, c}, {a, b, c}}$

∩

{

}

\cap \mathscr{A} = \{ a \}

$\cap A = {a}$

11、 ... and 、 Set operation priority

The first kind of operation ( Monocular operator ) : Absolutely make up , Power set , Generalized intersection , Generalized Union ; Operations are performed from left to right ;

The second kind of operation ( Binocular operator ) : Primary union , Primary delivery , Relative complement , Symmetry difference ; Operate in the order of parentheses , Without parentheses, operations are performed from left to right ;

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