# 不交并

## 定义与记法

${\displaystyle I}$为一个指标集，${\displaystyle \{A_{i};\;i\in I\}}$是一个集合族，则

${\displaystyle \bigcup _{i\in I}A_{i}}$

${\displaystyle A_{i}\cap A_{j}=\varnothing }$[1]:1

${\displaystyle \bigsqcup _{i\in I}A_{i}}$

${\displaystyle \sum _{i\in I}A_{i}}$

${\displaystyle \bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\{(x,i):x\in A_{i}\}.}$

## 任意集合族的不交并

${\displaystyle I}$为一个指标集，${\displaystyle \{A_{i};\;i\in I\}}$是一个集合族，则首先定义：

${\displaystyle \forall i\in I,A_{i}^{*}=\{(i,x);\;x\in A_{i}\}}$

${\displaystyle \bigcup _{i\in I}A_{i}^{*}}$

### 例子

${\displaystyle A_{1}^{*}=\{(1,x);\;x\in A_{1}\}=\{(1,{\frac {1}{2}})\}}$
${\displaystyle A_{2}^{*}=\{(2,x);\;x\in A_{2}\}=\{(2,{\frac {1}{4}}),(2,{\frac {1}{2}}),(2,{\frac {3}{4}})\}}$
${\displaystyle A_{3}^{*}=\{(3,x);\;x\in A_{3}\}=\{(3,{\frac {1}{8}}),(3,{\frac {1}{4}}),(3,{\frac {3}{8}}),(3,{\frac {1}{2}}),(3,{\frac {5}{8}}),(3,{\frac {3}{4}}),(3,{\frac {7}{8}})\}}$

${\displaystyle \bigsqcup _{i\in \mathbb {Z} ^{+}}A_{i}^{*}=\{(i,{\frac {j}{2^{i}}});\;\;(i,j)\in \mathbb {Z} ^{+}\times \mathbb {Z} ^{+},j<2^{i}\}}$

${\displaystyle \bigsqcup _{i\in \mathbb {Z} ^{+}}A_{i}=\{(i,{\frac {j}{2^{i}}});\;\;(i,j)\in \mathbb {Z} ^{+}\times \mathbb {Z} ^{+},j<2^{i}\}}$${\displaystyle \bigcup _{i\in \mathbb {Z} ^{+}}^{*}A_{i}=\{(i,{\frac {j}{2^{i}}});\;\;(i,j)\in \mathbb {Z} ^{+}\times \mathbb {Z} ^{+},j<2^{i}\}}$

## 参考来源

1. ^ E. Artin. Geometric Algebra. John Wiley & Sons. 2011. ISBN 978-1-118-16454-9.
2. ^ Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs. The Grammar of Graphics. Springer. 2006. ISBN 978-0-387-28695-2.
3. ^ Lang Serge. Algebra. Springer. 2005. ISBN 978-2-10-007980-3.