mirror of
https://github.com/Rushilwiz/math381.git
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finished 5.1 of hw4
This commit is contained in:
parent
2d5e7e7ae6
commit
90b86c9451
13
homework
13
homework
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@ -2,9 +2,18 @@
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from string import Template
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import os
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import sys
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import datetime
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def next_weekday(d, weekday):
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days_ahead = weekday - d.weekday()
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if days_ahead <= 0: # Target day already happened this week
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days_ahead += 7
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return d + datetime.timedelta(days_ahead)
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i = input("What number homework is it? ")
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d = input("What is the due date (e.g. September 12)? ")
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d = input("Is there a special due date? (if not leave empty) (e.g. September 12)? ")
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if d == "": d = next_weekday(datetime.datetime.now(), 3).strftime("%B %-d")
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s = {
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'number': i,
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@ -16,4 +25,4 @@ with open(f'hw/.template/template.tex', 'r') as f:
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result = src.substitute(s)
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os.mkdir(f'hw/hw{i}')
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with open(f'hw/hw{i}/hw{i}.tex', 'w') as f:
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f.write(result)
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f.write(result)
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4
hw/hw4/hw4.aux
Normal file
4
hw/hw4/hw4.aux
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@ -0,0 +1,4 @@
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\relax
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\@writefile{toc}{\contentsline {section}{Unit 5.1}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{Unit 2.1}{5}{}\protected@file@percent }
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\gdef \@abspage@last{11}
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157
hw/hw4/hw4.fdb_latexmk
Normal file
157
hw/hw4/hw4.fdb_latexmk
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@ -0,0 +1,157 @@
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# Fdb version 4
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["pdflatex"] 1696363543 "/home/rushil/school/math/hw/hw4/hw4.tex" "hw4.pdf" "hw4" 1696363544 0
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|
||||
\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}
|
||||
Loading…
Reference in New Issue
Block a user