# 區間

## 定义

### 实区间

${\displaystyle (a,b)=\{x\in \mathbb {R} \colon a
${\displaystyle [a,b]=\{x\in \mathbb {R} \colon a\leq x\leq b\}}$

${\displaystyle (a,b]=\{x\in \mathbb {R} \colon a
${\displaystyle [a,b)=\{x\in \mathbb {R} \colon a\leq x

${\displaystyle (a,\infty )=\{x\in \mathbb {R} \colon x>a\},}$
${\displaystyle [a,\infty )=\{x\in \mathbb {R} \colon x\geq a\},}$

${\displaystyle (-\infty ,b)=\{x\in \mathbb {R} \colon x
${\displaystyle (-\infty ,b]=\{x\in \mathbb {R} \colon x\leq b\},}$

${\displaystyle (-\infty ,\infty )=\mathbb {R} }$

### 偏序集或预序集中的区间

${\displaystyle (a,b)=\{x\in X\colon a
${\displaystyle [a,b]=\{x\in X\colon a\lesssim x\lesssim b\}}$
${\displaystyle (a,b]=\{x\in X\colon a
${\displaystyle [a,b)=\{x\in X\colon a\lesssim x
${\displaystyle (a,\infty )=\{x\in X\colon a
${\displaystyle [a,\infty )=\{x\in X\colon a\lesssim x\}}$
${\displaystyle (-\infty ,b)=\{x\in X\colon x
${\displaystyle (-\infty ,b]=\{x\in X\colon x\lesssim b\}}$
${\displaystyle (-\infty ,\infty )=X}$

${\displaystyle {\bar {X}}=X\sqcup \{-\infty ,\infty \}}$
${\displaystyle -\infty

### 序凸集和序凸分支

${\displaystyle \mathbb {Q} =\{x\in \mathbb {Q} \colon x^{2}<2\}}$

${\displaystyle (X,\lesssim )}$ 是一个预序集，且${\displaystyle Y\subseteq X}$ 。包含在${\displaystyle Y}$ 中的${\displaystyle X}$ 的序凸集关于包含关系构成偏序集。这个偏序集的极大元叫做${\displaystyle Y}$ 序凸分支[3]:Definition 5.1佐恩引理，包含在${\displaystyle Y}$ 中的${\displaystyle X}$ 的任意序凸集包含于${\displaystyle Y}$ 的一个序凸分支，然而这种序凸分支不一定是唯一的。在全序集中，这样的序凸分支确实唯一。也就是说，全序集的子集的序凸分支构成分划

## 區間算術

${\displaystyle T\times S=\{x\mid {}}$ 屬於${\displaystyle T}$ 的某些${\displaystyle y}$ ，及屬於${\displaystyle S}$ 的某些${\displaystyle z}$ ，使得${\displaystyle x=y\times z\}}$

${\displaystyle [a,b]+[c,d]=[a+c,b+d]}$
${\displaystyle [a,b]-[c,d]=[a-d,b-c]}$
${\displaystyle [a,b]\times [c,d]=[\min\{ac,ad,bc,bd\},\max\{ac,ad,bc,bd\}]}$
${\displaystyle {\frac {[a,b]}{[c,d]}}=\left[\min \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\},\max \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\}\right]}$

## 另一種寫法

${\displaystyle \left]a,b\right[=\{x\mid a
${\displaystyle \left[a,b\right]=\{x\mid a\leq x\leq b\}}$
${\displaystyle \left[a,b\right[=\{x\mid a\leq x
${\displaystyle \left]a,b\right]=\{x\mid a

## 參考

1. ^ Interval and segment - Encyclopedia of Mathematics. encyclopediaofmath.org. Springer & The European Mathematical Society. [2021-05-18]. （原始内容存档于2014-12-26）.
2. ^ Vind, Karl. Independence, additivity, uncertainty. Studies in Economic Theory 14. Berlin: Springer. 2003. ISBN 978-3-540-41683-8. Zbl 1080.91001. doi:10.1007/978-3-540-24757-9 （英语）.
3. ^ Heath, R. W.; Lutzer, David J.; Zenor, P. L. Monotonically normal spaces. Transactions of the American Mathematical Society. 1973, 178: 481–493. ISSN 0002-9947. MR 0372826. Zbl 0269.54009. doi:10.2307/1996713 （英语）.
4. ^ ISO 31-11:1992. ISO. [2021-05-18]. （原始内容存档于2021-05-18） （英语）.