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Commit 148b5f5

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hengxin
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+1-13-boolean-algebra: +hw
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% 1-13-boolean-algebra.tex
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%%%%%%%%%%%%%%%%%%%%
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\documentclass[a4paper, justified]{tufte-handout}
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\input{hw-preamble} % feel free to modify this file
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%%%%%%%%%%%%%%%%%%%%
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\title{第13讲: 布尔代数}
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\me{魏恒峰}{hfwei@nju.edu.cn}{}{}
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\date{\zhtoday} % or like 2019年9月13日
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%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%
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\noplagiarism % always keep this line
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%%%%%%%%%%%%%%%%%%%%
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\begin{abstract}
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\mfigcap{width = 0.85\textwidth}{figs/George-Boole}{George Boole}
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% \begin{center}{\fcolorbox{blue}{yellow!60}{\parbox{0.65\textwidth}{\large
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% \begin{itemize}
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% \item
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% \end{itemize}}}}
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% \end{center}
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\end{abstract}
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%%%%%%%%%%%%%%%%%%%%
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\beginrequired
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%%%%%%%%%%%%%%%
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\begin{problem}[Definition]
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请证明: A bounded, distributive, and complemented lattice is a Boolean algebra.
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\end{problem}
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\begin{solution}
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\end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%
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\begin{problem}[$D_{n}$]
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请证明: $D_{n}$ (定义见阅读材料 Example 15.1 (c))
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是 Boolean algebra 当且仅当 $n = p_1 p_2 \cdots p_k$ (for some $k$),
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这里 $p_i$ 皆为素数且互异。
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\end{problem}
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\begin{solution}
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\end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%
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\begin{problem}[Atom]
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设 $B$ 为 Boolean algebra, 对于任意元素 $a \in B$,
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定义 $\textsf{Atom}(a) = \set{x \le a \mid x \text{ is an atom}}$。
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\noindent 现假设 $B$ 为有穷 Boolean algebra。
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请证明:
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\[
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\forall a \in B: a \neq 0 \implies \textsf{Atom}(a) \neq \emptyset.
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\]
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\end{problem}
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\begin{solution}
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\end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%
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\begin{problem}[Isomorphic]
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请证明: 有穷且等势的 Boolean algebras 均同构。
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\end{problem}
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\begin{solution}
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\end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
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\beginoptional
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%%%%%%%%%%%%%%%
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\begin{problem}[Isomorphic]
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是否任何 Boolean Algebra 都与某个幂集 Boolean Algebra 同构?
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请证明或给出反例。
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\end{problem}
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\begin{solution}
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\end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
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\beginot
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%%%%%%%%%%%%%%%
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\begin{ot}[]
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\end{ot}
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% \begin{solution}
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% \end{solution}
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%%%%%%%%%%%%%%%
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\vspace{0.50cm}
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%%%%%%%%%%%%%%%
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\begin{ot}[]
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\end{ot}
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% \begin{solution}
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% \end{solution}
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
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% 如果没有需要订正的题目,可以把这部分删掉
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% \begincorrection
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%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
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% 如果没有反馈,可以把这部分删掉
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\beginfb
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% 你可以写
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% ~\footnote{优先推荐 \href{problemoverflow.top}{ProblemOverflow}}:
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% \begin{itemize}
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% \item 对课程及教师的建议与意见
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% \item 教材中不理解的内容
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% \item 希望深入了解的内容
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% \item $\cdots$
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% \end{itemize}
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%%%%%%%%%%%%%%%%%%%%
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\end{document}

‎2019/1-13-boolean-algebra/1-13-boolean-algebra.tex.latexmain‎

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‎2019/1-13-boolean-algebra/README.md‎

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# README.md
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## Homework
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### OLD
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- 证明布尔代数是有界有补分配格,有界有补分配格是布尔代数
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- 证明SM定理15.6
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- 证明等势(有穷)的布尔代数均同构
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### NEW
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- 选做题:
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- 幂集布尔代数
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## OT
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### OLD
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- 卡诺图
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### NEW
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% hw-preamble.tex
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% geometry for A4 paper
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% See https://tex.stackexchange.com/a/119912/23098
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\geometry{
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left=20.0mm,
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top=20.0mm,
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bottom=20.0mm,
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textwidth=130mm, % main text block
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marginparsep=5.0mm, % gutter between main text block and margin notes
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marginparwidth=50.0mm % width of margin notes
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}
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% for colors
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\usepackage{xcolor} % usage: \color{red}{text}
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% predefined colors
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\newcommand{\red}[1]{\textcolor{red}{#1}} % usage: \red{text}
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\newcommand{\blue}[1]{\textcolor{blue}{#1}}
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\newcommand{\teal}[1]{\textcolor{teal}{#1}}
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\usepackage{todonotes}
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% heading
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\usepackage{sectsty}
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\setcounter{secnumdepth}{2}
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\allsectionsfont{\centering\huge\rmfamily}
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% for Chinese
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\usepackage{xeCJK}
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\usepackage{zhnumber}
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\setCJKmainfont[BoldFont=FandolSong-Bold.otf]{FandolSong-Regular.otf}
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% for fonts
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\usepackage{fontspec}
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\newcommand{\song}{\CJKfamily{song}}
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\newcommand{\kai}{\CJKfamily{kai}}
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% To fix the ``MakeTextLowerCase'' bug:
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% See https://github.com/Tufte-LaTeX/tufte-latex/issues/64#issuecomment-78572017
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% Set up the spacing using fontspec features
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\renewcommand\allcapsspacing[1]{{\addfontfeature{LetterSpace=15}#1}}
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\renewcommand\smallcapsspacing[1]{{\addfontfeature{LetterSpace=10}#1}}
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% for url
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\usepackage{hyperref}
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\hypersetup{colorlinks = true,
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linkcolor = teal,
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urlcolor = teal,
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citecolor = blue,
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anchorcolor = blue}
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\newcommand{\me}[4]{
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\author{
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{\bfseries 姓名:}\underline{#1}\hspace{2em}
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{\bfseries 学号:}\underline{#2}\hspace{2em}\\[10pt]
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{\bfseries 评分:}\underline{#3\hspace{3em}}\hspace{2em}
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{\bfseries 评阅:}\underline{#4\hspace{3em}}
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}
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}
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% Please ALWAYS Keep This.
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\newcommand{\noplagiarism}{
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\begin{center}
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\fbox{\begin{tabular}{@{}c@{}}
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请独立完成作业,不得抄袭。\\
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若得到他人帮助, 请致谢。\\
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若参考了其它资料,请给出引用。\\
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鼓励讨论,但需独立书写解题过程。
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\end{tabular}}
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\end{center}
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}
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\newcommand{\goal}[1]{
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\begin{center}{\fcolorbox{blue}{yellow!60}{\parbox{0.50\textwidth}{\large
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\begin{itemize}
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\item 体会``思维的乐趣''
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\item 初步了解递归与数学归纳法
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\item 初步接触算法概念与问题下界概念
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\end{itemize}}}}
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\end{center}
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}
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% Each hw consists of four parts:
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\newcommand{\beginrequired}{\hspace{5em}\section{作业 (必做部分)}}
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\newcommand{\beginoptional}{\section{作业 (选做部分)}}
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\newcommand{\beginot}{\section{Open Topics}}
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\newcommand{\begincorrection}{\section{订正}}
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\newcommand{\beginfb}{\section{反馈}}
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% for math
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\usepackage{amsmath, mathtools, amsfonts, amssymb}
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\newcommand{\set}[1]{\{#1\}}
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% define theorem-like environments
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\usepackage[amsmath, thmmarks]{ntheorem}
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\theoremstyle{break}
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\theorempreskip{2.0\topsep}
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\theorembodyfont{\song}
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\theoremseparator{}
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\newtheorem{problem}{题目}[subsection]
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\renewcommand{\theproblem}{\arabic{problem}}
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\newtheorem{ot}{Open Topics}
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\theorempreskip{3.0\topsep}
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\theoremheaderfont{\kai\bfseries}
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\theoremseparator{:}
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\theorempostwork{\bigskip\hrule}
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\newtheorem*{solution}{解答}
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\theorempostwork{\bigskip\hrule}
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\newtheorem*{revision}{订正}
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\theoremstyle{plain}
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\newtheorem*{cause}{错因分析}
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\newtheorem*{remark}{注}
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\theoremstyle{break}
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\theorempostwork{\bigskip\hrule}
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\theoremsymbol{\ensuremath{\Box}}
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\newtheorem*{proof}{证明}
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% \newcommand{\ot}{\blue{\bf [OT]}}
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% for figs
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\renewcommand\figurename{图}
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\renewcommand\tablename{表}
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% for fig without caption: #1: width/size; #2: fig file
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\newcommand{\fig}[2]{
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\begin{figure}[htbp]
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\centering
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\includegraphics[#1]{#2}
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}
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% for fig with caption: #1: width/size; #2: fig file; #3: caption
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\newcommand{\figcap}[3]{
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% for fig with both caption and label: #1: width/size; #2: fig file; #3: caption; #4: label
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\newcommand{\figcaplbl}[4]{
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\centering
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% for margin fig without caption: #1: width/size; #2: fig file
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\newcommand{\mfig}[2]{
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% for margin fig with caption: #1: width/size; #2: fig file; #3: caption
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}
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\usepackage{fancyvrb}
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% for algorithms
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\usepackage[]{algorithm}
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\usepackage[]{algpseudocode} % noend
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% See [Adjust the indentation whithin the algorithmicx-package when a line is broken](https://tex.stackexchange.com/a/68540/23098)
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\newcommand{\algparbox}[1]{\parbox[t]{\dimexpr\linewidth-\algorithmicindent}{#1\strut}}
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\newcommand{\hStatex}[0]{\vspace{5pt}}
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\makeatletter
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\newlength{\trianglerightwidth}
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\settowidth{\trianglerightwidth}{$\triangleright$~}
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\algnewcommand{\LineComment}[1]{\Statex \hskip\ALG@thistlm \(\triangleright\) #1}
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\parbox[t]{\dimexpr\linewidth-\ALG@thistlm}{\hangindent=\trianglerightwidth \hangafter=1 \strut$\triangleright$ #1\strut}}
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\makeatother
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% see https://tex.stackexchange.com/a/133265/23098
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\usepackage{tikz}
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\renewcommand\thefootnote{\protect\circled{\arabic{footnote}}}

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