从原课件中做的笔记使用课程语言 即英文 其余来源笔记使用中文 便于读者与作者区分

Learning Outcomes

  • Understand and use notation for sets and subsets.
  • Recognize and use well-known finite and infinite sets.
  • Understand and use the operrations of set union, intersections, and difference.
  • Represent set operations using Venn diagrams.

Basic Definitions

Well-known inginite sets

Symbol Meaning
$\mathbb{N}$ The set of positive integers
$\mathbb{Z}$ The set of all integers
$\mathbb{Q}$ The set of rational numbers $\frac{a}{b}$, where $a \in \mathbb{Z}$ and $b \in \mathbb{N}$
$\mathbb{R}$ The set of all real number
$\mathbb{C}$ The set of complex numbers $a+b$ $i$ , where $a,b\in \mathbb{R}$ and $i = \sqrt{-1}$

1.1

A set is a collection of distinct objects that is thought of as a whole. The objects in a set are called elements or members of the set.

$a\in A$ : the object $a$ is a member of the set $A$.
$a\notin A$ : the object $a$ is not a member of the set $A$.
$A=$ {$1,2,3,4$} : is a set expressed as a finite list of obkect in curly braces.

1.2

Let $A$ and $B$ be sets. We say that $A$ is a subset of $B$ if every element of $A$ is an element of $B$, and we denote this by writing $A \subset B$

Sometimes, the notation $A \subseteq B$ is used to denote that $A$ is a subset of $B$. (It’s a little bit wrong.) Then $A \subseteq B$ explicitly states that $A$ is a proper subset of $B$. (There is at least one element of $B$ which is not in $A$)

1.3

Let $A$ and $B$ be sets. We say that $A = B$, denoted by $A = B$ if $A \subset B & B \subset A$.

Basic Theorem

1.1

Transitive Law of subsets If $A \subset B$ and $B \subset C$, then $A \subset C$.

集合序列的上极限和下极限

设{ ${A_n,n \ge 1}$ }，求$\bigcap\limits_{n=1}^\infty \bigcup\limits_{k=n}^\infty A_k$ 及$\bigcup\limits_{n=1}^\infty \bigcap\limits_{k=n}^\infty A_k$，
其中$A_1=${ $1,a$ }，$A_2=${ $0,b$ }，$A_3=${ $1,b$ }，$A_4=${ $0,b$ }，$A_5=${ $1,b$ }。

Here ：

① $\bigcap\limits_{n=1}^\infty \bigcup\limits_{k=n}^\infty A_k$ 的解法是先并再交。
当$n=1$时，记$P_1=\bigcup\limits_{k=n=1}^\infty A_k=${ $0,1,a,b$ }，
当$n=2$时，记$P_2=\bigcup\limits_{k=n=2}^\infty A_k=${ $0,1,b$ }，

所以，$\bigcap\limits_{n=1}^\infty \bigcup\limits_{k=n}^\infty A_k=P_1\bigcap P_2\bigcap P_3\bigcap\cdots=${ $0,1,b$ }
若记$\lim\limits_{n\to\infty}supA_n=\bigcap\limits_{n=1}^\infty \bigcup\limits_{k=n}^\infty A_k$
则，$\lim\limits_{n\to\infty}supA_n=$ {$w|w$属于无穷多个$A_n$}

② $\bigcup\limits_{n=1}^\infty \bigcap\limits_{k=n}^\infty A_k$ 的解法是先交再并。
当$n=1$时，记$Q_1=\bigcap\limits_{k=n=1}^\infty A_k=\varnothing$，
当$n=2$时，记$Q_2=\bigcap\limits_{k=n=2}^\infty A_k=${ $b$ }，

所以，$\bigcup\limits_{n=1}^\infty \bigcap\limits_{k=n}^\infty A_k=Q_1\bigcap Q_2\bigcap Q_3\bigcap\cdots=$ { $b$ }
若记$\lim\limits_{n\to\infty}inf A_n=\bigcup\limits_{n=1}^\infty \bigcap\limits_{k=n}^\infty A_k$
则，$\lim\limits_{n\to\infty}inf A_n=$ {$w|w$至多不属于有限多个$A_n$}