Add theorem commands

MA1511/sets.tex

@@ -254,11 +254,11 @@ $+$ and $\times$ on $\Z/k$:
 \section*{Countable Sets}
 A set $X$ is finite if $\exists n \geq 0. $ a bijection $\{1, ...n\} \to X$ \\
 Pigeonhole Principle: for finite $X$, any injective $f: X \to X$ is also surjective. \\
-$\N$ is infinite. Proof: $f: \N \to \N$ is trivially injective, and $\neg\exists x.~f(x) = 0$, and so not surjective. \\
+$\N$ is infinite. Proof: $f: \N \to \N $ defined by $ f(x) = x + 1$ is trivially injective, and $\neg\exists x.~f(x) = 0$, and so not surjective. \\
 By the inverse of the Pigeonhole Principle, $\N$ is infinite.\\
 A set $X$ is \emph{countably infinite} iff there exists a bijection $\N \to X$. \\
 A set is \emph{countable} iff it is finite or countably infinite. \\
-Any subset of $\N$ is countable. Proof: Let $X \in \N$.\\
+Any subset of $\N$ is countable. Proof: Let $X \subseteq \N$.\\
 If $X$ is finite, it's trivially countable.
 Otherwise, $X$ is infinite and it must be shown that $X$ is countably infinite. \\
 For $k \in \N$, $X_{>k} = \{ n \in X | n > k \}$. Then $X_{>k} \not= \varnothing$, as $X$ would be a subset of $\{1..k\}$ and would be finite. \\

MA2008/2024-09-24

@@ -0,0 +1 @@
+

MA2008/decls.tex

@@ -0,0 +1,76 @@
+\documentclass[fleqn]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage[margin=0.25in]{geometry}
+\usepackage{enumitem}
+\usepackage{systeme}
+\usepackage{mathtools}
+
+\date{}
+\author{}
+
+\newcommand{\paren}[1]{\left(#1\right)}
+\newcommand{\powerset}[1]{\mathcal{P}\paren{#1}}
+\renewcommand{\Re}[1]{\operatorname{\mathbb{R}e}\paren{#1}}
+\renewcommand{\Im}[1]{\operatorname{\mathbb{{I}}m}\paren{#1}}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\Q}{\mathbb{Q}}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\conj}[1]{\overline{#1}}
+\renewcommand{\mod}[1]{\left|#1\right|}
+\newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}}
+\newcommand{\adj}[1]{\operatorname{adj}#1}
+\newcommand{\card}[1]{\left|#1\right|}
+\newcommand{\littletaller}{\mathchoice{\vphantom{\big|}}{}{}{}}
+\newcommand{\restr}[2]{{% we make the whole thing an ordinary symbol
+  \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
+  #1 % the function
+  \littletaller % pretend it's a little taller at normal size
+  \right|_{#2} % this is the delimiter
+  }}
+
+\makeatletter
+\renewcommand*{\env@matrix}[1][*\c@MaxMatrixCols c]{%
+  \hskip -\arraycolsep
+  \let\@ifnextchar\new@ifnextchar
+  \array{#1}}
+\makeatother
+
+
+% https://gitlab.com/jim.hefferon/linear-algebra/-/blob/master/src/sty/linalgjh.sty
+\newlength{\grsteplength}
+\setlength{\grsteplength}{1.5ex plus .1ex minus .1ex}
+
+\newcommand{\grstep}[2][\relax]{%
+   \ensuremath{\mathrel{
+       \hspace{\grsteplength}\mathop{\longrightarrow}\limits^{#2\mathstrut}_{
+                                     \begin{subarray}{l} #1 \end{subarray}}\hspace{\grsteplength}}}}
+\newcommand{\repeatedgrstep}[2][\relax]{\hspace{-\grsteplength}\grstep[#1]{#2}}
+
+\newcommand{\swap}{\leftrightarrow}
+
+% https://tex.stackexchange.com/a/198806
+\makeatletter
+\newcommand{\subalign}[1]{%
+  \vcenter{%
+    \Let@ \restore@math@cr \default@tag
+    \baselineskip\fontdimen10 \scriptfont\tw@
+    \advance\baselineskip\fontdimen12 \scriptfont\tw@
+    \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+    \lineskiplimit\lineskip
+    \ialign{\hfil$\m@th\scriptstyle##$&$\m@th\scriptstyle{}##$\hfil\crcr
+      #1\crcr
+    }%
+  }%
+}
+\makeatother
+
+\theoremstyle{definition}
+\newtheorem*{theorem}{Theorem}
+\newtheorem*{lemma}{Lemma}
+\newtheorem*{corollary}{Corollary}
+
+\theoremstyle{remark}
+\newtheorem*{note}{Note}

MA2008/linear-transforms.tex

@@ -0,0 +1,27 @@
+\input{decls.tex}
+\title{Vector Spaces and Linear Transformations}
+\begin{document}
+\maketitle
+
+\begin{description}
+\item[Linear Transformation] A function $\phi: V \to W$ between vector spaces $V$ and $W$ (over some field $K$), such that
+	\begin{align*}
+		\phi(v + w) & \equiv \phi(v) + \phi(w) \\
+		\phi(x \cdot v) & \equiv x \cdot \phi(v) \tag{For $x \in K$}
+	\end{align*}
+\end{description}
+
+Differentiation is a linear transformation.
+Solutions to $f'' + f = 0$ for function $f$ are a vector space.
+
+\begin{theorem}
+For any scalars $\lambda, \mu \in \R$, there is a unique solution such that $f(0) = \mu$ and $f'(0) = \lambda$
+\end{theorem}
+The vector space is then two-dimensional, with basis $sin(x), cos(x)$
+
+
+\subsection*{}
+Vector spaces are used over finite fields in \emph{Algebraic Coding Theory}. The field is $\mathbb{F}_2 = \{0, 1\}$ - the integers mod 2.
+Binary strings of length $n$ are then a vector space over $\mathbb{F}_2^n$.
+ECC can be based on vector subspaces of $F_2^n$. (Vector subspaces are closed subsets of a vector space).
+\end{document}

MA2008/tmpl.tex

@@ -0,0 +1,6 @@
+\input{decls.tex}
+\title{}
+\begin{document}
+\maketitle
+
+\end{document}