From eed72a101a6c65221829bc5fe6ce9601290cc4f5 Mon Sep 17 00:00:00 2001 From: Rushil Umaretiya Date: Wed, 15 Nov 2023 16:42:08 -0500 Subject: [PATCH] finished hw8 --- hw/hw8/hw8.aux | 4 + hw/hw8/hw8.fdb_latexmk | 158 +++++++++++ hw/hw8/hw8.fls | 619 +++++++++++++++++++++++++++++++++++++++++ hw/hw8/hw8.log | 498 +++++++++++++++++++++++++++++++++ hw/hw8/hw8.pdf | Bin 0 -> 140306 bytes hw/hw8/hw8.synctex.gz | Bin 0 -> 20595 bytes hw/hw8/hw8.tex | 271 ++++++++++++++++++ 7 files changed, 1550 insertions(+) create mode 100644 hw/hw8/hw8.aux create mode 100644 hw/hw8/hw8.fdb_latexmk create mode 100644 hw/hw8/hw8.fls create mode 100644 hw/hw8/hw8.log create mode 100644 hw/hw8/hw8.pdf create mode 100644 hw/hw8/hw8.synctex.gz create mode 100644 hw/hw8/hw8.tex diff --git a/hw/hw8/hw8.aux b/hw/hw8/hw8.aux new file mode 100644 index 0000000..68c731f --- /dev/null +++ b/hw/hw8/hw8.aux @@ -0,0 +1,4 @@ +\relax +\@writefile{toc}{\contentsline {section}{Unit 4.1}{2}{}\protected@file@percent } 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+\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 8} +\newcommand{\hmwkDueDate}{November 16, 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}} + +\renewcommand{\mod}{\textbf{ mod }} +\renewcommand{\pmod}[1]{\textbf{ (mod }#1)} + +\begin{document} + +\maketitle + +\pagebreak + +\unit{4.1} +\prob{10} +Prove that if \(a\) and \(b\) are nonzero integers, \(a | b\), and \(a + b\) is odd, then \(a\) is odd. +\begin{proof} + Direct proof. Assume \(a\) and \(b\) are nonzero integers, \(a | b\), and \(a + b\) is odd. Then, by definition of divisibility, \(\exists k \in \AllIntegers, ak = b\). We also know that \(a + b\) is odd, so \(\exists j \in \AllIntegers, 2j + 1 = a + b\). Substituting \(b\) for \(ak\), we get \(2j + 1 = a + ak\). Factoring out \(a\), we get \(2j + 1 = a(1 + k)\). Since \(1 + k\) is an integer, we know that \(a\) is odd. +\end{proof} +\pagebreak +\prob{26} +Evaluate these quantites: +\begin{align*} +\end{align*} +\begin{enumerate}[a)] + \item \(-17 \textbf{ mod } 2\) = \hspace{12pt} 1 + \item \(144 \textbf{ mod } 7\) = \hspace{14pt} 4 + \item \(-101 \textbf{ mod } 13\) = \hspace{5pt}3 + \item \(199 \textbf{ mod } 19\) = \hspace{9pt} 9 +\end{enumerate} +\pagebreak +\prob{34} +Decide whether each of these integers is congruent to 3 modulo 7. +\begin{enumerate}[a)] + \item 37 + \begin{align*} + 7 \nmid (37 - 3)\\ + 37 \not\equiv 3 \pmod{7} + \end{align*} + \item 66 + \begin{align*} + 7 \mid (66 - 3)\\ + 66 \equiv 3 \pmod{7} + \end{align*} + \item -17 + \begin{align*} + 7 \nmid (-17 - 3)\\ + -17 \not\equiv 3 \pmod{7} + \end{align*} + \item -67 + \begin{align*} + 7 \mid (-67 - 3)\\ + -67 \equiv 3 \pmod{7} + \end{align*} +\end{enumerate} + +\pagebreak +\prob{36} +Find each of these values. + +\textbf{Note: } The following rules exist for modular arithmetic where \(a, b, n \in \AllIntegers, n \geq 2\): +\begin{align*} + (a + b) \textbf{ mod } n &= ((a \textbf{ mod } n) + (b \textbf{ mod } n)) \textbf{ mod } n & \text{Addition Law}\\ + (a \cdot b) \textbf{ mod } n &= ((a \textbf{ mod } n) \cdot (b \textbf{ mod } n)) \textbf{ mod } n & \text{Multiplication Law}\\ +\end{align*} + +\begin{enumerate}[a)] + \item \((177 \mod 31 + 270 \mod 31) \mod 31\) + \begin{align*} + (177 \mod 31 + 270 \mod 31) \mod 31 &= (177 + 270) \mod 31\\ + &= 447 \mod 31\\ + &= 13 + \end{align*} + \item \((177 \mod 31 \cdot 270 \mod 31) \mod 31\) + \begin{align*} + (177 \mod 31 \cdot 270 \mod 31) \mod 31 &= (177 \cdot 270) \mod 31\\ + &= 47790 \mod 31\\ + &= 19 + \end{align*} +\end{enumerate} +\pagebreak +\unit{6.1} +\prob{8} +How many diļ¬€erent three-letter initials with none of the letters repeated can people have? + +\begin{align*} + 26 \cdot 25 \cdot 24 = 15600 +\end{align*} + +\pagebreak + +\prob{22} +How many positive integers less than 1000, + +\begin{enumerate}[a)] + \item are divisible by 7? + \begin{align*} + \lfloor \frac{1000}{7} \rfloor = 142 + \end{align*} + \item are divisible by 7 but not by 11? + \begin{align*} + \lfloor \frac{1000}{7} \rfloor - \lfloor \frac{1000}{77} \rfloor = 142 - 12 = 130 + \end{align*} + \item are divisible by both 7 and 11? + \begin{align*} + \lfloor \frac{1000}{77} \rfloor = 12 + \end{align*} + \item are divisible by either 7 or 11? + \begin{align*} + \lfloor \frac{1000}{7} \rfloor + \lfloor \frac{1000}{11} \rfloor - \lfloor \frac{1000}{77} \rfloor = 142 + 90 - 12 = 220 + \end{align*} + \item are divisible by exactly one of 7 and 11? + \begin{align*} + \lfloor \frac{1000}{7} \rfloor + \lfloor \frac{1000}{11} \rfloor - 2 \lfloor \frac{1000}{77} \rfloor = 142 + 90 - 24 = 208 + \end{align*} + \item are divisible by neither 7 nor 11? + \begin{align} + 999 - 220 = 779 + \end{align} + \item have distinct digits? + \begin{align*} + 9 && \text{One digit}\\ + (10-1) \cdot 9 && \text{Two digits}\\ + (10-1) \cdot 9 \cdot 8 && \text{Three digits}\\ + 9 + 9 \cdot 9 + 9 \cdot 8 \cdot 7 = 738 && \text{Total} + \end{align*} + \item have distinct digits and are even + \begin{align*} + 4 && \text{One digit}\\ + 9 + 8 \cdot 4 && \text{Two digits}\\ + 9 \cdot 8 + 8 \cdot 8 \cdot 4 && \text{Three digits}\\ + 4 + (9 + 8 \cdot 4) + (9 \cdot 8 + 8 \cdot 8 \cdot 4) = 373 && \text{Total} + \end{align*} + +\end{enumerate} +\pagebreak +\prob{28} +How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits? + +\begin{align*} + 10^3 \cdot 26^3 + 26^3 \cdot 10^3 = 3.5152 \cdot 10^7 +\end{align*} +\pagebreak +\end{document} \ No newline at end of file