From f49b69dcab3172222dba96a829df5bfe9d0c0bef Mon Sep 17 00:00:00 2001 From: Rushil Umaretiya Date: Mon, 30 Oct 2023 02:16:24 -0400 Subject: [PATCH] hw7 --- hw/hw7/hw7.aux | 4 + hw/hw7/hw7.fdb_latexmk | 158 +++++++++++ hw/hw7/hw7.fls | 622 +++++++++++++++++++++++++++++++++++++++++ hw/hw7/hw7.log | 510 +++++++++++++++++++++++++++++++++ hw/hw7/hw7.pdf | Bin 0 -> 151472 bytes hw/hw7/hw7.synctex.gz | Bin 0 -> 23613 bytes hw/hw7/hw7.tex | 277 ++++++++++++++++++ 7 files changed, 1571 insertions(+) create mode 100644 hw/hw7/hw7.aux create mode 100644 hw/hw7/hw7.fdb_latexmk create mode 100644 hw/hw7/hw7.fls create mode 100644 hw/hw7/hw7.log create mode 100644 hw/hw7/hw7.pdf create mode 100644 hw/hw7/hw7.synctex.gz create mode 100644 hw/hw7/hw7.tex diff --git a/hw/hw7/hw7.aux b/hw/hw7/hw7.aux new file mode 100644 index 0000000..6f06153 --- /dev/null +++ b/hw/hw7/hw7.aux @@ -0,0 +1,4 @@ +\relax +\@writefile{toc}{\contentsline {section}{Unit 2.3}{2}{}\protected@file@percent } 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0000000..9b8af8a --- /dev/null +++ b/hw/hw7/hw7.tex @@ -0,0 +1,277 @@ +\documentclass{article} + +\usepackage{fancyhdr} +\usepackage{extramarks} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amsfonts} +\usepackage{tikz} +\usepackage[plain]{algorithm} +\usepackage{algpseudocode} +\usepackage[shortlabels]{enumitem} +\usepackage{mathtools} +\usepackage{amssymb} + +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ (\hmwkClassInstructor\ \hmwkClassTime): \hmwkTitle} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newcommand{\hmwkTitle}{Homework 7} +\newcommand{\hmwkDueDate}{November 2, 2023} +\newcommand{\hmwkClass}{Discrete Math} +\newcommand{\hmwkClassTime}{Section 003} +\newcommand{\hmwkClassInstructor}{Reese Lance} +\newcommand{\hmwkAuthorName}{\textbf{Rushil Umaretiya}} + +% +% Title Page +% + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\ + \normalsize\vspace{0.1in}\small{\textbf{Due\ on\ \hmwkDueDate\ at 11:59pm}}\\ + \normalsize\text{Tuesday/Thursday 11:00-12:15, Phillips 383}\\ + \vspace{0.1in}\large{\textit{\hmwkClassInstructor\ - \hmwkClassTime}} + \vspace{3in} +} + +\author{\hmwkAuthorName\\\small{rumareti@unc.edu}} +\date{} + +\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large Solution}} + +\newcommand{\unit}[1]{\section{Unit #1}} +\newcommand{\problem}[1]{\textbf{\##1}} +\newcommand{\prob}[1]{\problem{#1}} + + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\renewcommand{\And}{\wedge} +\newcommand{\Or}{\vee} +\newcommand{\Xor}{\oplus} +\newcommand{\Not}{\neg} +\newcommand{\Implies}{\rightarrow} +\newcommand{\Iff}{\leftrightarrow} +\newcommand{\union}{\cup} +\newcommand{\intersection}{\cap} + +\newcommand{\AllIntegers}{\mathbb{Z}} +\newcommand{\AllNaturals}{\mathbb{N}} +\newcommand{\AllRationals}{\mathbb{Q}} +\newcommand{\AllReals}{\mathbb{R}} +\newcommand{\AllComplexes}{\mathbb{C}} + +\begin{document} + +\maketitle + + +\pagebreak + +\unit{2.3} +\prob{44} +Let \(f\) be the function from \(\AllReals\) to \(\AllReals\) defined by +\(f(x) = x^2\). Find + +\begin{enumerate}[a)] + \item \(f^{-1}(\{1\})\) + \item \(f^{-1}(\{x \mid 0 < x < 1\})\) + \item \(f^{-1}(\{x \mid x > 4\})\) +\end{enumerate} +\textbf{Note: } \(f^{-1}(x) = \{\pm\sqrt{x}\}\) +\begin{enumerate}[a)] + \item \{1, -1\} + \item \(\{x \neq 0 \mid -1 < x < 1\}\) + \item \(\{x \mid (x > 2) \Or (x < -2)\}\) +\end{enumerate} + +\pagebreak + +\prob{46} +Let \(f\) be a function from \(A\) to \(B\). Let \(S\) and \(T\) be subsets of \(B\). Show that + +\begin{enumerate}[a)] + \item \(f^{-1}(S \union T) = f^{-1}(S) \union f^{-1}(T)\) + \begin{proof}We will use the definition of \(f^{-1}(S)\) to prove this. + \begin{enumerate} + \item[\(\rightarrow\)] Let \(x \in f^{-1}(S \union T)\). Then \(f(x) \in S \union T\). Thus \(f(x) \in S\) or \(f(x) \in T\). Thus \(x \in f^{-1}(S)\) or \(x \in f^{-1}(T)\). Thus \(x \in f^{-1}(S) \union f^{-1}(T)\). + \item[\(\leftarrow\)] Let \(x \in f^{-1}(S) \union f^{-1}(T)\). Then \(x \in f^{-1}(S)\) or \(x \in f^{-1}(T)\). Thus \(f(x) \in S\) or \(f(x) \in T\). Thus \(f(x) \in S \union T\). Thus \(x \in f^{-1}(S \union T)\). + \end{enumerate} + Therefore, \(f^{-1}(S \union T) = f^{-1}(S) \union f^{-1}(T)\). + \end{proof} + \item \(f^{-1}(S \intersection T) = f^{-1}(S) \intersection f^{-1}(T)\) + \begin{proof} + We will use the same method as part a) to prove this. + \begin{enumerate} + \item [\(\rightarrow\)] Let \(x \in f^{-1}(S \intersection T)\). Then \(f(x) \in S \intersection T\). Thus \(f(x) \in S\) and \(f(x) \in T\). Thus \(x \in f^{-1}(S)\) and \(x \in f^{-1}(T)\). Thus \(x \in f^{-1}(S) \intersection f^{-1}(T)\). + \item [\(\leftarrow\)] Let \(x \in f^{-1}(S) \intersection f^{-1}(T)\). Then \(x \in f^{-1}(S)\) and \(x \in f^{-1}(T)\). Thus \(f(x) \in S\) and \(f(x) \in T\). Thus \(f(x) \in S \intersection T\). Thus \(x \in f^{-1}(S \intersection T)\). + \end{enumerate} + Therefore, \(f^{-1}(S \intersection T) = f^{-1}(S) \intersection f^{-1}(T)\). + \end{proof} +\end{enumerate} +\pagebreak + + +\unit{4.1} +\prob{12} +Prove that if \(a\) is a positive integer, then \(4\) does not divide \(a^2 + 2\). +\begin{proof} + We will conduct a proof by contradicition. Assume \(4 \mid a^2 + 2\). This means that \(\exists k \in \AllIntegers, 4k = a^2 + 2\). We can rewrite this as \(a^2 = 4k - 2 = 2(2k - 1)\). This means that \(a^2\) is even which means that \(a\) is even. This means that \(\exists j \in \AllIntegers, a = 2j\). We can rewrite this as \(a^2 = 4j^2\). This means that \(a^2\) is divisible by 4. This means that \(4 \mid a^2\). This means that \(4 \mid a^2 + 2\) and \(4 \mid a^2\). We can then use identites to show that \(4 \mid a^2 - (a^2 + 2) \equiv 4 \mid 2\). This is a contradiction because \(4 \nmid 2\). Therefore, \(4 \nmid a^2 + 2\). +\end{proof} + +\pagebreak + +\prob{30} +Find the integer a such that\\ +\begin{enumerate}[a)] + \item \(a \equiv 43\pmod{23}\) and \(-22 \leq a \leq 0\) + \begin{align*} + a = -3 + \end{align*} + \item \(a \equiv 17\pmod{29}\) and \(-14 \leq a \leq 14\) + \begin{align*} + a = -12 + \end{align*} + \item \(a \equiv -11\pmod{21}\) and \(90 \leq a \leq 110\) + \begin{align*} + a = 94 + \end{align*} +\end{enumerate} +\pagebreak + +\prob{34} +Decide whether each of these integers is congruent to 3 modulo 7.\\ +\textbf{Note: } \(a \equiv b \pmod n\) means that \(n \mid (a - b)\). +\begin{enumerate}[a)] + \item 37 + \begin{align*} + 37 \not \equiv 3 \pmod 7 \text{ because } 7 \nmid (37 - 3) = 34 + \end{align*} + \item 66 + \begin{align*} + 66 \equiv 3 \pmod 7 \text{ because } 7 \mid (66 - 3) = 63 + \end{align*} + \item -17 + \begin{align*} + -17 \not \equiv 3 \pmod 7 \text{ because } 7 \nmid (-17 - 3) = -20 + \end{align*} + \item -67 + \begin{align*} + -67 \equiv 3 \pmod 7 \text{ because } 7 \mid (-67 - 3) = -70 + \end{align*} +\end{enumerate} +\pagebreak + +\prob{44} +Show that if \(n\) is an integer then \(n^2 \equiv 0\) or 1\(\pmod 4\). +\begin{proof} + We will conduct a proof by cases on \(n\). + \begin{enumerate} + \item [\(n\) is even] + \begin{align*} + n = 2k \text{ for some } k \in \AllIntegers\\ + n^2 = 4k^2 = 4(k^2)\\ + \exists \bar{k} \in \AllIntegers, k^2 = \bar{k}\\ + n^2 = 4\bar{k}\\ + n^2 \equiv 0 \pmod 4 + \end{align*} + \item [\(n\) is odd] + \begin{align*} + n = 2k + 1 \text{ for some } k \in \AllIntegers\\ + n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1\\ + \exists \bar{k} \in \AllIntegers, k^2 + k = \bar{k}\\ + n^2 = 4\bar{k} + 1\\ + n^2 \equiv 1 \pmod 4 + \end{align*} + \end{enumerate} + Therefore, \(n^2 \equiv 0\) or 1\(\pmod 4\). +\end{proof} +\pagebreak + +\prob{46} +Prove that if \(n\) is an odd positive integer, then \(n^2 \equiv 1 \pmod 8\). +\begin{proof} + We will conduct a direct proof. Let \(n\) be an odd positive integer. + \begin{align*} + \exists k \in \AllIntegers, n = 2k + 1\\ + n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1 = 4(k)(k+1) + 1\\ + \end{align*} + Note, either \(k\) or \(k+1\) must be even, therefore + \begin{align*} + \exists \bar{k} \in \AllIntegers, k(k+1) = 2\bar{k}\\ + n^2 = 4(2\bar{k}) + 1 = 8\bar{k} + 1\\ + n^2 \equiv 1 \pmod 8 + \end{align*} +\end{proof} +\pagebreak + + +\end{document} \ No newline at end of file