diff --git a/homework b/homework index 8f4238a..24419c6 100755 --- a/homework +++ b/homework @@ -2,9 +2,18 @@ from string import Template import os import sys +import datetime + +def next_weekday(d, weekday): + days_ahead = weekday - d.weekday() + if days_ahead <= 0: # Target day already happened this week + days_ahead += 7 + return d + datetime.timedelta(days_ahead) i = input("What number homework is it? ") -d = input("What is the due date (e.g. September 12)? ") +d = input("Is there a special due date? (if not leave empty) (e.g. September 12)? ") + +if d == "": d = next_weekday(datetime.datetime.now(), 3).strftime("%B %-d") s = { 'number': i, @@ -16,4 +25,4 @@ with open(f'hw/.template/template.tex', 'r') as f: result = src.substitute(s) os.mkdir(f'hw/hw{i}') with open(f'hw/hw{i}/hw{i}.tex', 'w') as f: - f.write(result) \ No newline at end of file + f.write(result) diff --git a/hw/hw4/hw4.aux b/hw/hw4/hw4.aux new file mode 100644 index 0000000..c4dcd01 --- /dev/null +++ b/hw/hw4/hw4.aux @@ -0,0 +1,4 @@ +\relax +\@writefile{toc}{\contentsline {section}{Unit 5.1}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{Unit 2.1}{5}{}\protected@file@percent } +\gdef \@abspage@last{11} diff --git a/hw/hw4/hw4.fdb_latexmk b/hw/hw4/hw4.fdb_latexmk new file mode 100644 index 0000000..ff840cb --- /dev/null +++ b/hw/hw4/hw4.fdb_latexmk @@ -0,0 +1,157 @@ +# Fdb version 4 +["pdflatex"] 1696363543 "/home/rushil/school/math/hw/hw4/hw4.tex" "hw4.pdf" "hw4" 1696363544 0 + "/etc/texmf/web2c/texmf.cnf" 1693368362 475 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+\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{} + +% +% Homework Problem Environment +% +% This environment takes an optional argument. When given, it will adjust the +% problem counter. This is useful for when the problems given for your +% assignment aren't sequential. See the last 3 problems of this template for an +% example. +% +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Problem \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkTitle}{Homework 4} +\newcommand{\hmwkDueDate}{October 5, 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{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{\AllIntegers}{\mathbb{Z}} +\newcommand{\AllRationals}{\mathbb{Q}} +\newcommand{\AllReals}{\mathbb{R}} +\newcommand{\AllComplexes}{\mathbb{C}} + +\begin{document} + +\maketitle + +\pagebreak + +\unit{5.1} +\prob{10} +\begin{enumerate}[a)] + \item Find a formula for + \begin{align*} + \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\ldots+\frac{1}{n(n+1)} + \end{align*} + by examining the values of this expression for small values of \(n\).\\ + +Given the following equation, +\begin{align*} + P(n) &= \sum_{i=1}^{n}\frac{1}{n(n+1)}\\ +\end{align*} +We can find the following values for \(P(x)\) and generalize a formula for \(P(n)\). +\begin{align*} + P(1) &&= \frac{1}{2}\\ + P(2) &= \frac{1}{2}+\frac{1}{6}&=\frac{2}{3}\\ + P(3) &= \frac{1}{2}+\frac{1}{6}+\frac{1}{12}&=\frac{3}{4}\\ + P(4) &= \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}&=\frac{4}{5}\\ + P(n) &&= \frac{n}{n+1} +\end{align*} + \item Prove the formula you conjectured in part (a). +\begin{proof} + We will prove the formula by induction.\\ + \textbf{Base Case:} \(n=1\) + \begin{align*} + P(1) = \frac{1}{1+1} = \frac{1}{2} + \end{align*} + \textbf{Inductive step:} + Given P(n), we will prove P(n+1). + \begin{align*} + P(n+1) &= \sum_{i=1}^{n+1}\frac{1}{n(n+1)}\\ + &= \sum_{i=1}^{n}\frac{1}{n(n+1)}+\frac{1}{(n+1)(n+2)}\\ + &= \frac{n}{n+1}+\frac{1}{(n+1)(n+2)}\\ + &= \frac{n(n+2)+1}{(n+1)(n+2)}\\ + &= \frac{(n+1)^2}{(n+1)(n+2)}\\ + &= \frac{n+1}{n+2} + \end{align*} +\end{proof} +\end{enumerate} +\pagebreak +\prob{34} +Prove that 6 divides \(n^3 - n\) whenever \(n\) is a nonnegative integer.\\ +\begin{proof} + Given \(P(n) = \exists k \in \AllIntegers, n^3-n = 6k\), we will show that \(\forall n \in \AllIntegers^+(P(n))\) by induction.\\ + \textbf{Base Case:} \(n=0\) + \begin{align*} + 0^3 - 0 = 6(0) + \end{align*} + \textbf{Inductive step:} + Given P(n), we will prove P(n+1). + \begin{align*}(n+1) + (n+1)^3-(n+1) &= n^3+3n^2+3n+1-(n+1)\\ + &= n^3+3n^2+2n\\ + &= n^3-n+(3n^2+3n)\\ + &= (n^3 - n) + 3(n)(n+1) + \end{align*} + Since we are assuming \(P(n)\), we can affirm that \(n^3-n\) is true. Now we can show that 6 also divides the second term, + \begin{align*} + \exists k \in \AllIntegers, 3(n)(n+1) &= 6k\\ + n(n+1) &= 2k + \end{align*} + Since any odd and even integer multiplied together is even, we can affirm that 6 divides \(3(n)(n+1)\). Since we know that, + \begin{align*} + \forall a,b,n \in \AllIntegers,\\n|a, n|b \implies n|(a+b). + \end{align*} + 6 must divide \((n^3 - n) + 3(n)(n+1)\). Thus \(P(n+1)\) is true. +\end{proof} + +\pagebreak +\prob{64} +Use mathematical induction to prove that if \(p\) is a prime and \(p | a_1a_2 \cdots a_n\), where \(a_i\) is an integer for \(i = 1, 2, 3, \dots , n\), then \(p | a_i\) for some integer \(i\). +\begin{align*} + P(n) = \forall a_1, a_2, \dots, a_n \in \AllIntegers, p | a_1a_2 \cdots a_n \implies p | a_i \text{ for some } i \in \AllIntegers .\\ +\end{align*} +\textbf{Base Case:} \(n=1\) +\begin{align*} + p | a_1 \implies p | a_1 +\end{align*} +\textbf{Inductive Step:} Given P(n), we will prove P(n+1). +\begin{align*} + P(n+1) &= p|a_1a_2\cdots a_na_{n+1} \implies p|a_i \text{ for some } i \in \AllIntegers .\\ +\end{align*} +Since we know that \(p\) is prime, +\begin{align*} + p|a_1a_2\cdots a_na_{n+1} &\implies p|a_1a_2\cdots a_n \text{ or } p|a_{n+1}\\ + &\implies P(n) \text{ or } p|a_{n+1}\\ + &\implies p|a_i \text{ for some } i \in \AllIntegers \text{ or } p|a_{n+1}\\ + &\implies P(n+1) +\end{align*} +We have shown that \(P(n) \implies P(n+1)\), thus \(P(n+1)\) is true. +\pagebreak +\unit{2.1} +\prob{2} Use set builder notation to give a description of each of these sets. +\begin{enumerate}[a)] + \item \{{0, 3, 6, 9, 12}\}\\ + \begin{align*} + A = \{x \in \AllIntegers^+ | x = 3n, 0 \leq n \leq 4\} + \end{align*} + \item \{{-3,-2,-1, 0, 1, 2, 3}\} + \begin{align*} + B = \{x \in \AllIntegers | -3 \leq x \leq 3\} + \end{align*} + \item \(\{m,n,o,p\}\) + \begin{align*} + C = \{x | x \text{ is a lowercase letter in the alphabet from m to p}\} + \end{align*} +\end{enumerate} +\pagebreak +\prob{6} For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. +\begin{enumerate}[a)] + \item the set of people who speak English, the set of people who speak English with an Australian accent + \item the set of fruits, the set of citrus fruits + \item the set of students studying discrete mathematics, the set of students studying data structures +\end{enumerate} +\pagebreak +\prob{12} Determine whether these statements are true or false. +\begin{enumerate}[a)] + \item \(\emptyset \in \{\emptyset\}\) + \item \(\emptyset \in \{\emptyset, \{\emptyset\}\}\) + \item \(\{\emptyset\} \in \{\emptyset\}\) + \item \(\{\emptyset\} \in \{\{\emptyset\}\}\) + \item \(\{\emptyset\} \subset \{\emptyset, \{\emptyset\}\}\) + \item \(\{\{\emptyset\}\} \subset \{\emptyset, \{\emptyset\}\}\) + \item \(\{\{\emptyset\}\} \subset \{\{\emptyset\}, \{\emptyset\}\}\) +\end{enumerate} +\pagebreak +\prob{18} +Use a Venn diagram to illustrate the relationships \(A \subset B\) and \(A \subset C\). + +\pagebreak +\prob{22} +What is the cardinality of each of these sets? +\begin{enumerate}[a)] + \item \(\emptyset\) + \item \(\{\emptyset\}\) + \item \(\{\emptyset, \{\emptyset\}\}\) + \item \(\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\) +\end{enumerate} + +\pagebreak +\prob{32} Suppose that \(A \times B = \emptyset\), where \(A \text{ and } B\) are sets. What can you conclude? + +\pagebreak +\prob{44} Prove or disprove that if \(A, B, \text{and } C\) are nonempty sets and \(A \times B = A \times C\text{, then } B = C\). + +\end{document} \ No newline at end of file