From 60d8c54b7a25dec57e0a077751d8d2927f23b6fa Mon Sep 17 00:00:00 2001 From: Jacob Strieb Date: Fri, 27 Mar 2020 14:50:52 -0400 Subject: [PATCH] Separate style/shortcuts into class and package --- Makefile | 1 + example.pdf | Bin 95317 -> 103247 bytes example.tex | 33 +++++++++++++++++++++-------- homework.cls | 53 ++++++++--------------------------------------- hw-shortcuts.sty | 37 +++++++++++++++++++++++++++++++++ 5 files changed, 71 insertions(+), 53 deletions(-) create mode 100644 hw-shortcuts.sty diff --git a/Makefile b/Makefile index a3431ca..592b137 100644 --- a/Makefile +++ b/Makefile @@ -5,5 +5,6 @@ LATEX-DIR := $$HOME/texmf/tex/latex/local .PHONY: install install: homework.cls mkdir -p $(LATEX-DIR) + cp hw-shortcuts.sty $(LATEX-DIR) cp homework.cls $(LATEX-DIR) @echo Homework template installed in $(LATEX-DIR)/homework.cls diff --git a/example.pdf b/example.pdf index b7d85f0d8ba443d4ca145bac1761828a438e6258..698bc242d6a1c66adb202700fab92edd5e58c45e 100644 GIT binary patch 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@@ -1,21 +1,24 @@ -\documentclass[name=Jacob\ Strieb, andrewid=jstrieb, course=69-420, num=8]{homework} +\documentclass[name=Jacob\ Strieb, andrewid=jstrieb, course=69-420, num=4]{homework} + +\usepackage{hw-shortcuts} \begin{document} -\problem{1} +\problem{1a} \begin{claim} The set of rational numbers ($\Q$) is countable. \end{claim} \begin{proof} - Let $x \in \Q$. Then, by definition, we can write $x = \frac{a}{b}$ for some $a, b \in \Z$. This fraction can be uniquely mapped to a tuple $(a, b) \in \Z \times \Z$. This means that there exists an injective mapping $\Q \to \Z \times \Z$ that maps each fraction to a tuple. Thus, we have: + Let $x \in \Q$. Then, by definition, we can write $x = \frac{a}{b}$ for some $a, b \in \Z, b \neq 0$. This fraction can be uniquely mapped to a tuple $(a, b) \in \Z \times \Z$. This means that there exists an injective mapping $\Q \to \Z \times \Z$ that maps each fraction to a tuple. Thus, we have: $$ |\Q| \leq |\Z \times \Z| \leq |\N| $$ - By the countability of $|\N|$, we have shown that $\Q$ is countable. + By the countability of $\N$, we have shown that $\Q$ is countable. \end{proof} -\newpage -\problem2 +\separator + +\problem{1b} Let $E$ be an event, and let $1_E : \Omega \to \{ 0, 1 \}$ be an indicator random variable such that: $$ 1_E(\omega) = \begin{cases} 1 & \omega \in E \\ @@ -36,7 +39,7 @@ \end{proof} \newpage -\problem3 +\problem2 \begin{claim} $e^{iy} = \cos y + i \sin y$ for $y \in \R$ @@ -55,6 +58,18 @@ \end{proof} \newpage +\problem{3} +\begin{claim} + $\P \neq \NP$ +\end{claim} + +\begin{proof} + We have already shown that $\P \subseteq \NP$, so to prove that $\NP \neq \P$ it suffices to show that $\NP \not \subseteq \P$. We begin by fixing a language $L \in \NP$. + + \todo % TODO: Complete this problem later +\end{proof} +\newpage + \problem4 \begin{claim} @@ -64,9 +79,9 @@ \begin{proof} First, we pick and order $k$ cards to be matched. The probability of the selected card orders matching is $\frac{1}{k!}$. - Now, we consider the probability that the remaining $n - k$ cards \textit{do not} match. Similar to the examples from class and the textbook, this is an instance of an indexed union of sets $A_i \in \mathcal{F}$ such that $A_i$ is the set of permutations in which $f(i) = i$. But since we are looking for the probability of this not happening, we consider the probability of the complement, computed as 1 minus the probability of this union of events. + Now, we consider the probability that the remaining $n - k$ cards \textit{do not} match. Similar to the examples from class and the textbook, this is an instance of an indexed union of sets $A_i \in \mathcal{F}$ such that $A_i$ is the set of permutations in which $f(i) = i$. But since we are looking for the probability of this not happening, we consider the probability of the complement, computed as $1$ minus the probability of this union of events. - By the Inclusion-Exclusion principle, based on the notes from class, we have that + By the Inclusion-Exclusion principle, based on the notes from class, we have that: $$ \prob{ \bigcup_{i = 1}^{n - k} A_i } = \sum_{i = 1}^{n - k} \frac{(-1)^i}{i!} $$ Putting everything together, we get the following, and we're done. diff --git a/homework.cls b/homework.cls index 519a93d..2737925 100644 --- a/homework.cls +++ b/homework.cls @@ -6,9 +6,9 @@ \LoadClass{article} -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Package options and document metadata -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \RequirePackage{xkeyval} @@ -46,9 +46,9 @@ \author{\name} -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Imports and formatting -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Set the page on letter paper with microtype improvements and better list formatting \RequirePackage[letterpaper]{geometry} @@ -79,9 +79,9 @@ \titlespacing{\section}{0em}{0em}{1em} -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Custom commands -- words, environments, and formatting -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Headers for numbered problems \newcommand{\problem}[1]{\section{Problem #1}} @@ -96,9 +96,9 @@ \newenvironment{claim}{\textit{Claim.}}{\vspace{-1em}} -%%%% -% Custom commands -- redefined -%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Custom commands -- redefined commands that should look or behave differently +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Redefine the ampersand to use the italic version by default - as-per Elements of Typographic Style % See: https://tex.stackexchange.com/a/47353/150811 @@ -121,38 +121,3 @@ % Redefine the bar and vec commands to use larger overlines and bold-faced vectors \renewcommand{\bar}{\overline} \renewcommand{\vec}{\mathbf} - - -%%%% -% Custom commands -- math symbols and math shortcuts -%%%% - -% Shortcuts for left/right parentheses, square brackets, and curly braces -\newcommand{\lrp}[1]{\left( #1 \right)} -\newcommand{\lrs}[1]{\left[ #1 \right]} -\newcommand{\lrc}[1]{\left{ #1 \right}} - -% Commonly-used sets -\newcommand{\N}{\ensuremath{\mathbb{N}}} -\newcommand{\Z}{\ensuremath{\mathbb{Z}}} -\newcommand{\R}{\ensuremath{\mathbb{R}}} -\newcommand{\Q}{\ensuremath{\mathbb{Q}}} -\newcommand{\C}{\ensuremath{\mathbb{C}}} - -% Powerset and shortcut for derivative -\newcommand{\powerset}{\ensuremath{\mathcal{P}}} -\newcommand{\ddx}{\ensuremath{\frac{d}{dx}}} - -% Probability measure, expectation, and variance formatting -\newcommand{\expect}[1]{\mathbb{E}\left[ #1 \right]} -\newcommand{\prob}[1]{\mathbb{P}\left[ #1 \right]} -\newcommand{\var}[1]{\mathrm{var}\left( #1 \right)} - -% Commonly-used computer science languages -\renewcommand{\P}{\ensuremath{\mathsf{P}}} -\newcommand{\NP}{\ensuremath{\mathsf{NP}}} -\newcommand{\coNP}{\ensuremath{\mathsf{coNP}}} - -% Commonly-used reductions add semantic clarity -\newcommand{\CookReducesTo}{\ensuremath{\leq^P_T}} -\newcommand{\KarpReducesTo}{\ensuremath{\leq^P_m}} \ No newline at end of file diff --git a/hw-shortcuts.sty b/hw-shortcuts.sty new file mode 100644 index 0000000..b69e30e --- /dev/null +++ b/hw-shortcuts.sty @@ -0,0 +1,37 @@ +% Required for TeX packages +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{hw-shortcuts}[2020/03/27 Custom homework shortcuts as a package] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Custom commands -- math symbols and math shortcuts +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Shortcuts for left/right parentheses, square brackets, and curly braces +\newcommand{\lrp}[1]{\left( #1 \right)} +\newcommand{\lrs}[1]{\left[ #1 \right]} +\newcommand{\lrc}[1]{\left{ #1 \right}} + +% Commonly-used sets +\newcommand{\N}{\ensuremath{\mathbb{N}}} +\newcommand{\Z}{\ensuremath{\mathbb{Z}}} +\newcommand{\R}{\ensuremath{\mathbb{R}}} +\newcommand{\Q}{\ensuremath{\mathbb{Q}}} +\newcommand{\C}{\ensuremath{\mathbb{C}}} + +% Powerset and shortcut for derivative +\newcommand{\powerset}{\ensuremath{\mathcal{P}}} +\newcommand{\ddx}{\ensuremath{\frac{d}{dx}}} + +% Probability measure, expectation, and variance formatting +\newcommand{\expect}[1]{\mathbb{E}\left[ #1 \right]} +\newcommand{\prob}[1]{\mathbb{P}\left[ #1 \right]} +\newcommand{\var}[1]{\mathrm{var}\left( #1 \right)} + +% Commonly-used computer science languages +\renewcommand{\P}{\ensuremath{\mathsf{P}}} +\newcommand{\NP}{\ensuremath{\mathsf{NP}}} +\newcommand{\coNP}{\ensuremath{\mathsf{coNP}}} + +% Commonly-used reductions add semantic clarity +\newcommand{\CookReducesTo}{\ensuremath{\leq^P_T}} +\newcommand{\KarpReducesTo}{\ensuremath{\leq^P_m}} \ No newline at end of file