集的笛卡尔积

术语“乘积”在数学上表示将两个或多个值相乘时获得的结果。例如，45是9和5的乘积。必须熟悉诸如联合和相交之类的集合的基本操作，它们要在2个或更多集合上执行。笛卡尔积也是在两个集合上执行的一个这样的操作，它返回一组有序对。

什么是有序对?

有序对是一对对象，其中一个元素首先分配，另一个元素第二个分配，用(a，b)表示。在这里，“ a”被称为有序集合的第一个组成部分，而“ b”被称为有序集合的第二个组成部分。

Example: (5, 7) is an ordered pair of integers. 

Note: (5, 7) ≠ (7, 5), an ordered pair (a, b) is equal to (x, y) only if a = x and b = y.

集的笛卡尔积

两个非空集合A和B的笛卡尔积是所有可能的有序对的集合，其中该对的第一个分量来自A，该对的第二个分量来自B。这样获得的有序对的集合为用A×B表示。

A×B = {(a，b)：a∈A和b∈B}

Example: 

Let A = {1, 2} and B = {4, 5, 6}

A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}
 

Here the first component of every ordered pair is from set A the second component is from set B.

两组的笛卡尔积可以很容易地以矩阵的形式表示，其中两组都在任一轴上，如下图所示。 A = {1，2}和B = {x，y，z}的笛卡尔积

笛卡尔积的性质

1.笛卡尔积是不可交换的：A×B≠B×A

Example: 

A = {1, 2} , B = {a, b}

A × B = {(1, a), (1, b), (2, a), (2, b)}

B × A = {(a, 1), (b, 1), (b, 1), (b, 2)}

Therefore as A ≠ B we have A × B ≠ B × A
 

2. A×B = B×A，仅当A = B时

Proof:

Let A × B = B × A then we have  

A ⊆  B  and B ⊆  A, it follows that A = B
 

3.笛卡尔积的基数定义为A×B中的元素数，并且等于两个集合的基数的乘积：| A×B |。 = | A | * | B |

Proof:

Let a ∈ A then the number of ordered pair (a, b) such that b ∈ B is |B|. 

Therefore we have have |B| choices for b for each a where a ∈ A therefor the number of element in A×B is |A| * |B|.
 

4.如果A = {∅}或B = {∅}，则A×B = {∅}

Proof: 

We know |{∅}| = 0.

Now we have |A × B| = |{∅}| = 0

As |A × B| = |A| * |B| , we get |A| * |B| = 0

Thus atleast one of |A| or |B| should be equal to 0

Hence either A = {∅} or B = {∅}
 

集的有序对和笛卡尔积的样本问题

问题1：找到给定(2x – y，25)=(15，2x + y)的x和y的值?

解决方案：

As we know from the property of ordered pairs, 2x – y = 15 and 25 = 2x + y.

Solving the linear equations we have x = 10 and y = 5.

问题2。给定A = {2，3，4，5}并且B = {4，16，23}，则a∈A，b∈B，求出有序对的集合，使得a ²

解决方案：

As 2² < 16 and 23, 3² < 16 and 23, 4² < 23  

We have the set of ordered pairs such that a² < b is {(2, 16), (2, 23), (3, 16), (2, 23), (4, 23)}

问题3.如果A = {9，10}并且B = {3，4，6}，找到A×B和| A×B |?

解决方案：

A × B = {(9, 3), (9, 4), (9, 6), (10, 3), (10, 4), (10, 6)}

|A × B| = |A| * |B| = 2 * 3 = 6

问题4.如果A×B = {(a，x)，(a，y)，(b，x)，(b，y)}，找到A和B?

解决方案：

We know A is the set of all first components in ordered pairs of A × B and 

B is the set of the second component in the ordered pair of A × B.

Therefore A = {a, b} and B = {x, y}

问题5.给定A×B有15个有序对，A有5个元素，请问B中元素的数目是多少?

解决方案：

We know |A × B| = |A| * |B|, 15 = 5 * |B|

Therefore B has 15 / 5 = 3 elements.