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运算法则(笛卡尔乘积)

运算法则（笛卡尔乘积）

数学中，两个集合X和Y的笛卡⼉积(Cartesian product)，⼜称直积，表⽰为X × Y，是其第⼀个对象是X的成员⽽第⼆个对象是Y的⼀个成员的所有可能的有序对。

假设集合A={a,b}，集合B={0,1,2}，则两个集合的笛卡尔积为{(a,0),(a,1),(a,2),(b,0),(b,1), (b,2)}。

类似的例⼦有，如果A表⽰某学校学⽣的集合，B表⽰该学校所有课程的集合，则A与B的笛卡尔积表⽰所有可能的选课情况。

Union:

· For sets A and B:

· written A ∪ B,

· elements that are in A, or B, or both,

· A ∪ B = { x : x ∈ A or x ∈ B }.

· So that:

· (x ∈ A ∪ B) ⇔ (x ∈ A) ∨ (x ∈ B).

Intersection:

· For sets A and B:

· written A ∩ B,

· elements that are in both A and B,

· A ∩ B = { x : x ∈ A and x ∈ B }.

· So that:

· (x ∈ A ∩ B) ⇔ (x ∈ A) ∧ (x ∈ B);

· Two sets A and B are disjoint if they have

no elements in common:

A ∩ B = ∅.

Difference operator:

· For sets A and B:

· A \ B denotes

· elements that are in A but not in B,

· A \ B = { x ∈ A : x ∉ B }.

虹膜识别系统· So that:

· (x ∈ A \ B) ⇔ (x ∈ A) ∧ (x ∉ B)

韩炼润滑油Powerset:

· For a set A:

· written P(A),

Elements of the powerset are the subsets of A,

· P(A) = { X : X ⊆ A }.

· So that:

· (Y ∈ P (A)) ⇔ (Y ⊆ A), and

检索消除· ∅ ∈ P (A), and

· A ∈ P (A).

Ordered Pairs:

· Pairs of first and second objects,

· eg：co-ordinates (x, y) of points in a plane,

· ordering means that (a, b) = (b, a) ⇒ a = b.

· Cartesian Product:

For sets A and B:

· written A x B,

· ordered pairs of elements from A and B,

· A x B = { (a, b) : a ∈ A and b ∈ B }.

平板艺术音响

Generalisations to many sets:

· Union:

· eg A ∪ B ∪ C, or ∪F ;

· Intersection:

· eg A ∩ B ∩ C, or ∩ F ;

· Because ∪ and ∩ are associative;· Cartesian product:

窗前的气球教学设计

· eg A x B x C

盐酸金刚烷胺

= { (a, b, c) : a ∈ A ∧ b ∈ B ∧ c ∈ C }.

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