1% Style helpers for use in Math M413 2 3\usepackage[leqno]{amsmath} 4\usepackage{amsfonts} 5\usepackage{amssymb} 6\usepackage{amsthm} 7\usepackage{latexsym} 8 9\numberwithin{equation}{section} 10 11\DeclareMathOperator{\suchthat}{s.t.} 12\DeclareMathOperator{\st}{\;|\;} 13\DeclareMathOperator{\lub}{lub} 14\DeclareMathOperator{\glb}{glb} 15\DeclareMathOperator{\C}{C} 16\DeclareMathOperator{\Rl}{R_L} 17\DeclareMathOperator{\Rm}{R_M} 18\DeclareMathOperator{\Poly}{\mathcal{P}} 19\DeclareMathOperator{\NPoly}{\mathcal{NP}} 20\DeclareMathOperator{\outdegree}{out-degree} 21\DeclareMathOperator{\indegree}{in-degree} 22 23% Abbreviations for Commonly used sets 24\newcommand{\field}[1]{\mathbb{#1}} 25\newcommand{\Q}{\field{Q}} % Rational numbers 26\newcommand{\N}{\field{N}} % Natural numbers 27\newcommand{\R}{\field{R}} % Real numbers 28\newcommand{\Z}{\field{Z}} % Integers 29\newcommand{\Zp}{\field{Z}^+} % Positive Integers 30 31% Theorem declarations 32\theoremstyle{plain} 33\newtheorem{theorem}{Theorem}[section] 34\newtheorem{lemma}{Lemma}[section] 35\newtheorem{correllary}{Correllary}[section] 36\newtheorem{proposition}{Proposition}[section] 37 38\theoremstyle{definition} 39\newtheorem{definition}{Definition}[section] 40\newtheorem{example}{Example}[section] 41 42\theoremstyle{remark} 43\newtheorem*{remark}{Remark} 44\newtheorem*{note}{Note} 45\newtheorem{case}{Case} 46 47 48% Meaningful abbreviations, etc 49\newcommand{\onlyif}{\Rightarrow} 50\newcommand{\thus}{\Longrightarrow} 51\newcommand{\contradicts}{\rightarrow\leftarrow} 52\newcommand{\Union}{\bigcup} 53\newcommand{\union}{\cup} 54\newcommand{\Intersect}{\bigcap} 55\newcommand{\intersect}{\cap} 56\newcommand{\setdiff}{\setminus} 57\newcommand{\divides}{|} 58\newcommand{\Land}{\;\bigwedge\;} 59\newcommand{\Lor}{\;\bigvee\;} 60\newcommand{\floor}[1]{\lfloor#1\rfloor} 61\newcommand{\inverse}[1]{\ensuremath\frac{1}{#1}} 62\newcommand{\abs}[1]{\ensuremath \lvert#1\rvert} 63\newcommand{\norm}[1]{\ensuremath \lVert#1\rVert} 64\newcommand{\inner}[2]{\ensuremath #1 \cdot #2} 65 66\newcommand{\cross}{\times} 67\newcommand{\setcomplement}[1]{\overline{#1}} 68\newcommand{\emptystring}{\epsilon} 69\newcommand{\concat}[2]{#1 \cdot #2} 70\newcommand{\pair}[2]{\langle #1, #2 \rangle} 71\newcommand{\closure}[1]{#1^*} 72\newcommand{\strlen}[1]{\abs{#1}} 73 74\newcommand{\produce}{\rightarrow} 75\newcommand{\derive}{\Rightarrow} 76\newcommand{\derives}{\stackrel{*}{\Rightarrow}} 77