| % Style helpers for use in Math M413 |
| |
| \usepackage[leqno]{amsmath} |
| \usepackage{amsfonts} |
| \usepackage{amssymb} |
| \usepackage{amsthm} |
| \usepackage{latexsym} |
| |
| \numberwithin{equation}{section} |
| |
| \DeclareMathOperator{\suchthat}{s.t.} |
| \DeclareMathOperator{\st}{\;|\;} |
| \DeclareMathOperator{\lub}{lub} |
| \DeclareMathOperator{\glb}{glb} |
| \DeclareMathOperator{\C}{C} |
| \DeclareMathOperator{\Rl}{R_L} |
| \DeclareMathOperator{\Rm}{R_M} |
| \DeclareMathOperator{\Poly}{\mathcal{P}} |
| \DeclareMathOperator{\NPoly}{\mathcal{NP}} |
| \DeclareMathOperator{\outdegree}{out-degree} |
| \DeclareMathOperator{\indegree}{in-degree} |
| |
| % Abbreviations for Commonly used sets |
| \newcommand{\field}[1]{\mathbb{#1}} |
| \newcommand{\Q}{\field{Q}} % Rational numbers |
| \newcommand{\N}{\field{N}} % Natural numbers |
| \newcommand{\R}{\field{R}} % Real numbers |
| \newcommand{\Z}{\field{Z}} % Integers |
| \newcommand{\Zp}{\field{Z}^+} % Positive Integers |
| |
| % Theorem declarations |
| \theoremstyle{plain} |
| \newtheorem{theorem}{Theorem}[section] |
| \newtheorem{lemma}{Lemma}[section] |
| \newtheorem{correllary}{Correllary}[section] |
| \newtheorem{proposition}{Proposition}[section] |
| |
| \theoremstyle{definition} |
| \newtheorem{definition}{Definition}[section] |
| \newtheorem{example}{Example}[section] |
| |
| \theoremstyle{remark} |
| \newtheorem*{remark}{Remark} |
| \newtheorem*{note}{Note} |
| \newtheorem{case}{Case} |
| |
| |
| % Meaningful abbreviations, etc |
| \newcommand{\onlyif}{\Rightarrow} |
| \newcommand{\thus}{\Longrightarrow} |
| \newcommand{\contradicts}{\rightarrow\leftarrow} |
| \newcommand{\Union}{\bigcup} |
| \newcommand{\union}{\cup} |
| \newcommand{\Intersect}{\bigcap} |
| \newcommand{\intersect}{\cap} |
| \newcommand{\setdiff}{\setminus} |
| \newcommand{\divides}{|} |
| \newcommand{\Land}{\;\bigwedge\;} |
| \newcommand{\Lor}{\;\bigvee\;} |
| \newcommand{\floor}[1]{\lfloor#1\rfloor} |
| \newcommand{\inverse}[1]{\ensuremath\frac{1}{#1}} |
| \newcommand{\abs}[1]{\ensuremath \lvert#1\rvert} |
| \newcommand{\norm}[1]{\ensuremath \lVert#1\rVert} |
| \newcommand{\inner}[2]{\ensuremath #1 \cdot #2} |
| |
| \newcommand{\cross}{\times} |
| \newcommand{\setcomplement}[1]{\overline{#1}} |
| \newcommand{\emptystring}{\epsilon} |
| \newcommand{\concat}[2]{#1 \cdot #2} |
| \newcommand{\pair}[2]{\langle #1, #2 \rangle} |
| \newcommand{\closure}[1]{#1^*} |
| \newcommand{\strlen}[1]{\abs{#1}} |
| |
| \newcommand{\produce}{\rightarrow} |
| \newcommand{\derive}{\Rightarrow} |
| \newcommand{\derives}{\stackrel{*}{\Rightarrow}} |