# 笛卡儿积

${\displaystyle X\times Y=\left\{\left(x,y\right)\mid x\in X\land y\in Y\right\}}$

## 笛卡儿积的性质

• 对于任意集合${\displaystyle A}$ ，根据定义有${\displaystyle A\times \varnothing =\varnothing \times A=\varnothing }$
• 一般来说笛卡儿积不满足交换律结合律
• 笛卡儿积对集合的满足分配律，即
${\displaystyle A\times (B\cup C)=(A\times B)\cup (A\times C)}$
${\displaystyle (B\cup C)\times A=(B\times A)\cup (C\times A)}$
${\displaystyle A\times (B\cap C)=(A\times B)\cap (A\times C)}$
${\displaystyle (B\cap C)\times A=(B\times A)\cap (C\times A)}$
${\displaystyle (A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)}$
• 若一個集合${\displaystyle A}$ 包含有無限多的元素，那這個集合對自身的笛卡爾積${\displaystyle A\times A}$ 有和${\displaystyle A}$ 一樣多的元素。

## 笛卡儿平方和n元乘积

${\displaystyle \prod _{i=1}^{n}X_{i}:=X_{1}\times \ldots \times X_{n}:=\{(x_{1},\ldots ,x_{n})\ |\ x_{1}\in X_{1}\;\land \;\ldots \;\land \;x_{n}\in X_{n}\}}$

## 无穷乘积

${\displaystyle I_{x}=\{x(1),\,x(2),\,\dots ,\,x(n)\}}$
${\displaystyle \{1,\,2,\,\dots ,\,n\}\,{\overset {x}{\cong }}\,I_{x}}$
${\displaystyle (\forall i\in I)[f(i)\in x(i)]}$

${\displaystyle (f(1),\,f(2),\,\dots ,\,f(n))\in \prod _{i=1}^{n}x(i)}$

${\displaystyle I\,{\overset {x}{\cong }}\,{\mathcal {X}}}$

${\displaystyle \prod _{x}{\mathcal {X}}:=\left\{f\,{\bigg |}\,\left(f:I\to \bigcup {\mathcal {X}}\right)\wedge (\forall i\in I)[f(i)\in x(i)]\right\}}$

${\displaystyle \pi _{j}:\prod _{x}{\mathcal {X}}\to x(j)}$ ${\displaystyle \left(\forall f\in \prod _{x}{\mathcal {X}}\right)[\pi _{j}(f)=f(j)]}$

${\displaystyle \prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} ^{\omega }=\mathbb {R} \times \mathbb {R} \times \ldots }$

“非空集合的任意非空搜集的笛卡儿积为非空”這一陳述等价于选择公理

## 函数的笛卡儿积

${\displaystyle (f\times g)(a,x)=(f(a),g(x))}$