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uploaded first 3 hw
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6
.vscode/settings.json
vendored
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.vscode/settings.json
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{
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"[python]": {
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"editor.defaultFormatter": "ms-python.black-formatter"
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},
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"python.formatting.provider": "none"
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}
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19
homework
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homework
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#!/usr/bin/env python3
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from string import Template
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import os
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import sys
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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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s = {
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'number': i,
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'date': d
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}
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with open(f'hw/.template/template.tex', 'r') as f:
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src = Template(f.read())
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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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hw/.template/template.aux
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hw/.template/template.aux
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\relax
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\@writefile{toc}{\contentsline {section}{Unit X.X}{2}{}\protected@file@percent }
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\gdef \@abspage@last{2}
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150
hw/.template/template.fdb_latexmk
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hw/.template/template.fdb_latexmk
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# Fdb version 4
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File: tikzlibraryautomata.code.tex 2023-01-15 v3.1.10 (3.1.10)
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File: tikzlibraryshapes.multipart.code.tex 2023-01-15 v3.1.10 (3.1.10)
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/home/rushil/school/math/hw/.template/template.tex:149: Extra }, or forgotten $.
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<recently read> \egroup
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l.149 \maketitle
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I've deleted a group-closing symbol because it seems to be
|
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spurious, as in `$x}$'. But perhaps the } is legitimate and
|
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you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
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/home/rushil/school/math/hw/.template/template.tex:149: Extra }, or forgotten $.
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<recently read> \egroup
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l.149 \maketitle
|
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I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
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deleted material, e.g., by typing `I$}'.
|
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|
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/home/rushil/school/math/hw/.template/template.tex:149: LaTeX Error: There's no line here to end.
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See the LaTeX manual or LaTeX Companion for explanation.
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Type H <return> for immediate help.
|
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...
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l.149 \maketitle
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Your command was ignored.
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Type I <command> <return> to replace it with another command,
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or <return> to continue without it.
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/home/rushil/school/math/hw/.template/template.tex:149: Missing $ inserted.
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<inserted text>
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$
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l.149 \maketitle
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I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
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/home/rushil/school/math/hw/.template/template.tex:149: Missing } inserted.
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<inserted text>
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l.149 \maketitle
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I've inserted something that you may have forgotten.
|
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(See the <inserted text> above.)
|
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With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
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/home/rushil/school/math/hw/.template/template.tex:149: Missing } inserted.
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<inserted text>
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}
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l.149 \maketitle
|
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I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
[1
|
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|
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{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
|
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/home/rushil/school/math/hw/.template/template.tex:156: Missing $ inserted.
|
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<inserted text>
|
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$
|
||||
l.156 \end{document}
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
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you left one out. Proceed, with fingers crossed.
|
||||
|
||||
[2] (./template.aux) )
|
||||
Here is how much of TeX's memory you used:
|
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15639 strings out of 476091
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35749 multiletter control sequences out of 15000+600000
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522703 words of font info for 72 fonts, out of 8000000 for 9000
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1141 hyphenation exceptions out of 8191
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102i,13n,104p,456b,304s stack positions out of 10000i,1000n,20000p,200000b,200000s
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</usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsl10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb>
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Output written on template.pdf (2 pages, 95984 bytes).
|
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|
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|
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BIN
hw/.template/template.pdf
Normal file
BIN
hw/.template/template.pdf
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BIN
hw/.template/template.synctex.gz
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hw/.template/template.synctex.gz
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hw/.template/template.tex
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hw/.template/template.tex
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|
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|||
\documentclass{article}
|
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|
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\usepackage{fancyhdr}
|
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\usepackage{extramarks}
|
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\usepackage{amsmath}
|
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\usepackage{amsthm}
|
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\usepackage{amsfonts}
|
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\usepackage{tikz}
|
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\usepackage[plain]{algorithm}
|
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\usepackage{algpseudocode}
|
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\usepackage[shortlabels]{enumitem}
|
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\usepackage{mathtools}
|
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\usepackage{amssymb}
|
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|
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\usetikzlibrary{automata,positioning}
|
||||
|
||||
%
|
||||
% Basic Document Settings
|
||||
%
|
||||
|
||||
\topmargin=-0.45in
|
||||
\evensidemargin=0in
|
||||
\oddsidemargin=0in
|
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\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{}
|
||||
}
|
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|
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\setcounter{secnumdepth}{0}
|
||||
\newcounter{partCounter}
|
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\newcounter{homeworkProblemCounter}
|
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\setcounter{homeworkProblemCounter}{1}
|
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\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 $number}
|
||||
\newcommand{\hmwkDueDate}{$date, 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{X.X}
|
||||
\problem{X}
|
||||
|
||||
\end{document}
|
||||
4
hw/hw1/hw1.aux
Normal file
4
hw/hw1/hw1.aux
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{Unit 1.1}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{Unit 1.3}{4}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{4}
|
||||
0
hw/hw1/hw1.bbl
Normal file
0
hw/hw1/hw1.bbl
Normal file
149
hw/hw1/hw1.fdb_latexmk
Normal file
149
hw/hw1/hw1.fdb_latexmk
Normal file
|
|
@ -0,0 +1,149 @@
|
|||
# Fdb version 4
|
||||
["pdflatex"] 1694145544 "/home/rushil/school/math/hw/hw1.tex" "hw1.pdf" "hw1" 1694145544 0
|
||||
"/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 ""
|
||||
"/home/rushil/school/math/hw/hw1.tex" 1694145543 6845 6f0a9d7413c773083ce767950272b7e1 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
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||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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}{
|
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|
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|
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|
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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\title{
|
||||
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|
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\textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\
|
||||
\normalsize\vspace{0.1in}\small{Tuesday/Thursday 11:00-12:15, Phillips 383}\\
|
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\vspace{0.1in}\large{\textit{\hmwkClassInstructor\ - \hmwkClassTime}}
|
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|
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}
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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% Alias for the Solution section header
|
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|
||||
|
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\newcommand{\unit}[1]{\section{Unit #1}}
|
||||
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|
||||
\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}}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\pagebreak
|
||||
|
||||
\unit{1.1}
|
||||
\problem{2}
|
||||
\begin{enumerate}[a)]
|
||||
\item Not declarative, a command.
|
||||
\item Not declarative, a question.
|
||||
\item Is a proposition, not true: there are black flies in Maine.
|
||||
\item Not declarative, truth value can change based on \emph{x}.
|
||||
\item Is a proposition, not true; the moon is not made of cheese.
|
||||
\item Not declarative, truth value can change based on \emph{n}.
|
||||
\end{enumerate}
|
||||
\problem{4}
|
||||
\begin{enumerate}[a)]
|
||||
\item Janice does not have more Facebook friends than Juan.
|
||||
\item Quincy is not smarter than Venkat.
|
||||
\item Zelda does not drive more miles to school than Paola.
|
||||
\item Briana does not sleep longer than Gloria.
|
||||
\end{enumerate}
|
||||
\problem{10}
|
||||
\\Let \(p\) and \(q\) be the propositions:
|
||||
\begin{itemize}
|
||||
\item[\(p\):]I bought a lottery ticket this week.
|
||||
\item[\(q\):] I won the million dollar jackpot.
|
||||
\end{itemize}
|
||||
Express each of these propositions as an English sentence.
|
||||
\begin{enumerate}[a)]
|
||||
\item I did not buy a lottery ticket this week.
|
||||
\item I bought a lottery ticket this week or I won the million dollar jackpot.
|
||||
\item If I bought a lottery ticket this week, then I won the million dollar jackpot.
|
||||
\item I bought a lottery ticket this week and I won the million dollar jackpot.
|
||||
\item I bought a lottery ticket this week if and only if I won the million dollar jackpot.
|
||||
\item If I did not buy a lottery ticket this week, then I didn't win the million dollar jackpot.
|
||||
\item I did not buy a lottery ticket this week and I did not win the million dollar jackpot.
|
||||
\item I did not buy a lottery ticket this week, or I did buy a lottery ticket this week and won the million dollar jackpot.
|
||||
\end{enumerate}
|
||||
|
||||
\problem{18}
|
||||
\begin{enumerate}[a)]
|
||||
\item Both equations are true, therefore True.
|
||||
\item One equation is false and one is true, therefore False.
|
||||
\item Both propositions are false, therefore True.
|
||||
\item One equation is false and one is true, therefore False.
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\problem{20}
|
||||
\begin{enumerate}[a)]
|
||||
\item Both conditions are false, therefore the statement is true.
|
||||
\item Both conditions are false, therefore the statement is true.
|
||||
\item Since \(p\) is true and \(q\) is false, the statement is false.
|
||||
\item Both conditions are true, therefore the statement is true.
|
||||
\end{enumerate}
|
||||
\problem{22}
|
||||
\begin{enumerate}[a)]
|
||||
\item Inclusive or, you need proficiency in either language or both.
|
||||
\item Exclusive or, you can have either soup or salad, but not both.
|
||||
\item Inclusive or, you need either form of identification or both.
|
||||
\item Exclusive or, luckily you cannot both publish and perish.
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\unit{1.3}
|
||||
\problem{4b}
|
||||
\begin{displaymath}
|
||||
\begin{array}{|c c c|c|c|}
|
||||
|
||||
p & q & r & (p \wedge q)\wedge r & p \wedge(q \wedge r) \\
|
||||
\hline
|
||||
T & T & T & T & T \\
|
||||
T & T & F & F & F \\
|
||||
T & F & T & F & F \\
|
||||
T & F & F & F & F \\
|
||||
F & T & T & F & F \\
|
||||
F & T & F & F & F \\
|
||||
F & F & T & F & F \\
|
||||
F & F & F & F & F
|
||||
\end{array}
|
||||
\end{displaymath}
|
||||
\\
|
||||
Since the columns are identical the law is true.
|
||||
|
||||
\problem{6}
|
||||
\problem{8}
|
||||
\textbf{a)}
|
||||
\textbf{b)}
|
||||
\prob{32}
|
||||
|
||||
|
||||
\pagebreak
|
||||
|
||||
|
||||
\end{document}
|
||||
37
hw/hw2/hw2.aux
Normal file
37
hw/hw2/hw2.aux
Normal file
|
|
@ -0,0 +1,37 @@
|
|||
\relax
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||||
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\@writefile{toc}{\contentsline {subsection}{\#16 d)}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{Unit 1.4}{4}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\#6}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#10}{5}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{a)}{5}{}\protected@file@percent }
|
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\@writefile{toc}{\contentsline {subsection}{c)}{5}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{e)}{5}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\#12}{6}{}\protected@file@percent }
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||||
\@writefile{toc}{\contentsline {subsection}{e)}{6}{}\protected@file@percent }
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||||
\@writefile{toc}{\contentsline {subsection}{f)}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{g)}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#24}{7}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{a)}{7}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{b)}{7}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{c)}{7}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{d)}{7}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{e)}{7}{}\protected@file@percent }
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||||
\@writefile{toc}{\contentsline {section}{Unit 1.5}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#6}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{a)}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{c)}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{e)}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{f)}{8}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#24}{9}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{c)}{9}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{d)}{9}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{Unit 1.6}{10}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#8}{10}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#10}{11}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{c)}{11}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{d)}{12}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{e)}{13}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\#24}{14}{}\protected@file@percent }
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||||
\gdef \@abspage@last{14}
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||||
148
hw/hw2/hw2.fdb_latexmk
Normal file
148
hw/hw2/hw2.fdb_latexmk
Normal file
|
|
@ -0,0 +1,148 @@
|
|||
# Fdb version 4
|
||||
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|
||||
\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}
|
||||
\rhead{\firstxmark}
|
||||
\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{}
|
||||
|
||||
%
|
||||
% 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 2}
|
||||
\newcommand{\hmwkDueDate}{September 12, 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]{\subsection{\##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}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\pagebreak
|
||||
|
||||
\unit{1.3}
|
||||
\problem{16 b)}
|
||||
Show that \([(p \rightarrow q) \And (q \rightarrow r)] \rightarrow (p \rightarrow r)\) is a tautology.
|
||||
\begin{align*}
|
||||
[(p \rightarrow q) \And (q \rightarrow r)] \rightarrow (p \rightarrow r) &\equiv [(\neg p \Or q) \And (\neg q \Or r)] \rightarrow (\neg p \Or r) && \text{Logical equivalence.}\\
|
||||
&\equiv \neg [(\neg p \Or q) \And (\neg q \Or r)] \Or (\neg p \Or r) && \text{Logical equivalence.}\\
|
||||
&\equiv [\neg(\neg p \Or q) \And \neg( \neg q \Or r)] \Or (\neg p \Or r) && \text{De Morgan's Law.}\\
|
||||
&\equiv [(\neg \neg p \And \neg q) \Or (\neg \neg q \And \neg r)] \Or (\neg p \Or r) && \text{De Morgan's Law.}\\
|
||||
&\equiv [(p \And \neg q) \Or (q \And \neg r)] \Or (\neg p \Or r) && \text{Double Negation.}\\
|
||||
&\equiv [(p \Or (q \And \neg r)) \And (\neg q \Or (q \And \neg r))] \Or (\neg p \Or r) && \text{Distributive Law.}\\
|
||||
&\equiv [((p \Or q) \And (p \Or \neg r)) \And ((\neg q \Or q) \And (\neg q \Or \neg r))] \Or (\neg p \Or r) && \text{Distributive Law.}\\
|
||||
&\equiv [((p \Or q) \And (p \Or \neg r)) \And (T \And (\neg q \Or \neg r))] \Or (\neg p \Or r) && \text{Negation Law.}\\
|
||||
&\equiv [((p \Or q) \And (p \Or \neg r)) \And (\neg q \Or \neg r)] \Or (\neg p \Or r) && \text{Identity Law.}\\
|
||||
&\equiv [((p \Or q) \And (\neg q \Or \neg r)) \And (p \Or \neg r)] \Or (\neg p \Or r) && \text{Commutative Law.}\\
|
||||
&\equiv [((p \Or q) \And (\neg q \Or \neg r)) \Or (\neg p \Or r)] \And [(p \Or \neg r) \Or (\neg p \Or r)] && \text{Distributive Law.}\\
|
||||
&\equiv [((p \Or q) \And (\neg q \Or \neg r)) \Or (\neg p \Or r)] \And [(r \Or \neg r) \Or (\neg p \Or p)] && \text{Associative Law.}\\
|
||||
&\equiv [((p \Or q) \And (\neg q \Or \neg r)) \Or (\neg p \Or r)] \And [T \Or T] && \text{Negation Law.}\\
|
||||
&\equiv [((p \Or q) \And (\neg q \Or \neg r)) \Or (\neg p \Or r)] \And T && \text{Domination Law.}\\
|
||||
&\equiv ((p \Or q) \And (\neg q \Or \neg r)) \Or (\neg p \Or r) && \text{Identity Law.}\\
|
||||
&\equiv [((p \Or q) \Or (\neg p \Or r)) \And ((\neg q \Or \neg r) \Or (\neg p \Or r))] && \text{Distributive Law.}\\
|
||||
&\equiv [((p \Or \neg p) \Or (q \Or r)) \And ((\neg q \Or \neg p) \Or (\neg r \Or r))] && \text{Associative Law.}\\
|
||||
&\equiv [T \Or (q \Or r)) \And ((\neg q \Or \neg p) \Or T)] && \text{Negation Law.}\\
|
||||
&\equiv T \And T && \text{Domination Law.}\\
|
||||
&\equiv T && \text{Identity Law.}\\
|
||||
\end{align*}
|
||||
\pagebreak
|
||||
\problem{16 d)}
|
||||
Show that \([(p \Or q) \And (p \rightarrow r) \And (q \rightarrow r)] \rightarrow r\) is a tautology.
|
||||
\begin{align*}
|
||||
[(p \Or q) \And (p \rightarrow r) \And (q \rightarrow r)] \rightarrow r &\equiv \neg[(p \Or q) \And (\neg p \Or r) \And (\neg q \Or r )] \Or r &&\text{Logical equivalence.}\\
|
||||
&\equiv[(\neg p \And \neg q) \Or (p \And \neg r) \Or (q \And \neg r)] \Or r &&\text{De Morgan's Law.}\\
|
||||
&\equiv[(\neg p \And \neg q) \Or ((p \Or q) \And \neg r)] \Or r &&\text{Distributive Law.}\\
|
||||
&\equiv[((\neg p \And \neg q) \Or (p \Or q))\And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Distributive Law.}\\
|
||||
&\equiv[(\neg p \Or (p \Or q)) \And (\neg q \Or (p \Or q)) \And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Distributive Law.}\\
|
||||
&\equiv[(\neg p \Or p \Or q) \And (\neg q \Or p \Or q) \And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Associative Law.}\\
|
||||
&\equiv[(T \Or q) \And (T \Or p) \And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Negation Law.}\\
|
||||
&\equiv[T \And T \And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Domination Law.}\\
|
||||
&\equiv[T \And ((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Domination Law.}\\
|
||||
&\equiv[((\neg p \And \neg q) \Or \neg r)] \Or r &&\text{Identity Law.}\\
|
||||
&\equiv(\neg p \And \neg q) \Or (\neg r \Or r) &&\text{Associative Law.}\\
|
||||
&\equiv(\neg p \And \neg q) \Or T &&\text{Negation Law.}\\
|
||||
&\equiv T &&\text{Domination Law.}\\
|
||||
\end{align*}
|
||||
\pagebreak
|
||||
\unit{1.4}
|
||||
\problem{6}
|
||||
Let \(N(x)\) be the statement “\(x\) has visited North Dakota,”
|
||||
where the domain consists of the students in your school.
|
||||
Express each of these quantifications in English.
|
||||
\begin{enumerate}[a)]
|
||||
\item There exists a student in my school who has visited North Dakota.
|
||||
\item Every student in my school has visited North Dakota.
|
||||
\item There does not exist a student in my school who has visited North Dakota.
|
||||
\item There exists a student in my school who has not visited North Dakota.
|
||||
\item Not every student in my school has visited North Dakota.
|
||||
\item Every student in my school has not visited North Dakota.
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\problem{10}
|
||||
Let \(C(x)\) be the statement “x has a cat,” let \(D(x)\) be the statement “x has a dog,” and let \(F(x)\) be the statement “x has a ferret.” Express each of these statements in terms of \(C(x)\), \(D(x)\), \(F(x)\), quantifiers, and logical connectives.
|
||||
Let the domain consist of all students in your class.
|
||||
\subsection{a)}
|
||||
A student in your class has a cat, a dog, and a ferret.
|
||||
\[
|
||||
\exists x(C(x) \And D(x) \And F(x))
|
||||
\]
|
||||
\subsection{c)}
|
||||
Some student in your class has a cat and a ferret, but not a dog.
|
||||
\[
|
||||
\exists x(C(x) \And F(x) \And \neg D(x))
|
||||
\]
|
||||
\subsection{e)}
|
||||
For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.
|
||||
\[
|
||||
\exists x(C(x)) \And \exists x(D(x)) \And \exists x(F(x))
|
||||
\]
|
||||
\pagebreak
|
||||
\problem{12}
|
||||
Let \(Q(x)\) be the statement “\(x + 1 > 2x.\)” If the domain consists of all integers, what are these truth values?
|
||||
\subsection{e)}
|
||||
False. Q(1) is false.
|
||||
\subsection{f)}
|
||||
True. Q(1) is false, so \(\exists x \neg Q(x)\).
|
||||
\subsection{g)}
|
||||
False. Q(0) is true, since there exists a Q(x) that is not true.
|
||||
\pagebreak
|
||||
\problem{24}
|
||||
Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people.
|
||||
|
||||
\textbf{Note:} \(C(x)\) = "x is in your class"
|
||||
|
||||
\subsection{a)}
|
||||
Everyone in your class has a cellular phone. Where \(P(x)\) is the statement "x has a cellular phone."
|
||||
\[
|
||||
\text{Out of everyone in my class, } \forall xP(x) \\
|
||||
\]
|
||||
\[
|
||||
\text{Out of everyone, } \forall x(C(x) \rightarrow P(x))
|
||||
\]
|
||||
|
||||
\subsection{b)}
|
||||
Somebody in your class has seen a foreign movie. Where \(F(x)\) is the statement "x has seen a foreign movie."
|
||||
\[
|
||||
\text{Out of everyone in my class, } \exists xF(x) \\
|
||||
\]
|
||||
\[
|
||||
\text{Out of everyone, } \exists x(C(x) \And F(x))
|
||||
\]
|
||||
\subsection{c)}
|
||||
There is a person in your class who cannot swim. Where \(S(x)\) is the statement "x can swim."
|
||||
\[
|
||||
\text{Out of everyone in my class, } \exists x \neg S(x) \\
|
||||
\]
|
||||
\[
|
||||
\text{Out of everyone, } \exists x(C(x) \And \neg S(x))
|
||||
\]
|
||||
\subsection{d)}
|
||||
All students in your class can solve quadratic equations. Where \(Q(x)\) is the statement "x can solve quadratic equations."
|
||||
\[
|
||||
\text{Out of everyone in my class, } \forall xQ(x) \\
|
||||
\]
|
||||
\[
|
||||
\text{Out of everyone, } \forall x(C(x) \rightarrow Q(x))
|
||||
\]
|
||||
\subsection{e)}
|
||||
Some student in your class does not want to be rich. Where \(R(x)\) is the statement "x wants to be rich."
|
||||
\[
|
||||
\text{Out of everyone in my class, } \exists x \neg R(x) \\
|
||||
\]
|
||||
\[
|
||||
\text{Out of everyone, } \exists x(C(x) \And \neg R(x))
|
||||
\]
|
||||
\pagebreak
|
||||
\unit{1.5}
|
||||
\problem{6}
|
||||
Let \(C(x, y)\) mean that student \(x\) is enrolled in class \(y\), where the domain for \(x\) consists of all students in your school and the domain for \(y\) consists of all classes being given at your school. Express each of these statements by a simple English sentence.
|
||||
|
||||
|
||||
\subsection{a)}
|
||||
\[C(Randy Goldberg, CS 252)\]
|
||||
Randy Goldberg is enrolled in CS 252.
|
||||
\subsection{c)}
|
||||
\[\exists yC(Carol Sitea, y)\]
|
||||
There exists a class, \(y\), that Carol Sitea is enrolled in.
|
||||
\subsection{e)}
|
||||
\[\exists x \exists y \forall z((x \neq y) \And (C(x, z) \rightarrow C(y, z)))\]
|
||||
There exists two students, \(x\) and \(y\), such that for all classes, \(z\), student \(x\) and \(y\), if \(x\) is enrolled in \(z\), then \(y\) is enrolled in \(z\).
|
||||
\subsection{f)}
|
||||
\[\exists x\exists y\forall z((x \neq y) \And (C(x, z) \iff C(y, z)))\]
|
||||
There exists two students, \(x\) and \(y\), such that for all classes, \(z\), where student \(x\) and student \(y\) are not the same student, student \(x\) is enrolled in \(z\) if and only if student \(y\) is enrolled in \(z\).
|
||||
\pagebreak
|
||||
\problem{24}
|
||||
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.
|
||||
\subsection{c)}
|
||||
\[\exists x\exists y(((x \leq 0) \And (y \leq 0)) \And (x - y > 0))\]
|
||||
There exists two non-positive real numbers such that their difference is positive.
|
||||
\subsection{d)}
|
||||
\[\forall x\forall y((x \neq 0) \And (y \neq 0) \iff (xy \neq 0))\]
|
||||
For any two real numbers, \(x\) and \(y\), if and only if \(x\) and \(y\) are not zero, then their product is not zero.
|
||||
\pagebreak
|
||||
\unit{1.6}
|
||||
\problem{8}
|
||||
What rules of inference are used in this argument? “No man is an island. Manhattan is an island. Therefore, Manhattan is not a man.”\\
|
||||
\textbf{Let us assume:}\\
|
||||
\(P(x) = "x \text{ is a man}"\)\\
|
||||
\(Q(x) = "x \text{ is an island}"\)\\
|
||||
\[
|
||||
\begin{tabular}{lcc}
|
||||
& \textbf{Step} & \textbf{Reason} \\
|
||||
1. & \(\forall x(\neg P(x) \rightarrow Q(x))\) & Premise. \\
|
||||
2. & Q(Manhattan) & Premise. \\
|
||||
3. & P(Manhattan) \(\rightarrow\neg\) Q(Manhattan) & Universal instantiation from (1). \\
|
||||
4. & \(\neg(\neg\)Q(Manhattan)) & Double negation from (2).\\
|
||||
5. & \(\neg\) P(Manhattan) & Modus tollens from (3) and (4).
|
||||
\end{tabular}
|
||||
\]
|
||||
\pagebreak
|
||||
\problem{10}
|
||||
For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.
|
||||
\subsection{c)}
|
||||
\textbf{Assume:}\\
|
||||
P(x) = "x is an insect."\\
|
||||
Q(x) = "x has six legs."\\
|
||||
R(x, y) = "x eats y."\\
|
||||
\[
|
||||
\begin{tabular}{lcc}
|
||||
& \textbf{Step} & \textbf{Reason} \\
|
||||
1. & \(\forall x(P(x) \implies Q(x))\) & Premise. \\
|
||||
2. & P(Dragonflies) & Premise. \\
|
||||
3. & \(\neg\)Q(Spiders) & Premise. \\
|
||||
4. & R(Spiders, Dragonflies) & Premise. \\
|
||||
5. & P(Dragonflies) \(\implies\) Q(Dragonflies) & Universal instantiation from (1). \\
|
||||
6. & P(Spiders) \(\implies\) Q(Spiders) & Universal instantiation from (1). \\
|
||||
7. & Q(Dragonflies) & Modus ponens from (2) and (5). \\
|
||||
8. & \(\neg\)P(Spiders) & Modus tollens from (3) and (6). \\
|
||||
\end{tabular}
|
||||
\]
|
||||
\pagebreak
|
||||
\subsection{d)}
|
||||
\textbf{Assume:}\\
|
||||
P(x) = "x is a student."\\
|
||||
Q(x) = "x has an internet account."\\
|
||||
\[
|
||||
\begin{tabular}{lcc}
|
||||
& \textbf{Step} & \textbf{Reason} \\
|
||||
1. & \(\forall x(P(x) \implies Q(x))\) & Premise. \\
|
||||
2. & \(\neg\)Q(Homer) & Premise. \\
|
||||
3. & Q(Maggie) & Premise. \\
|
||||
4. & P(Homer) \(\implies\) Q(Homer) & Universal instantiation from (1). \\
|
||||
5. & \(\neg\)P(Homer) & Modus tollens from (2) and (4). \\
|
||||
\end{tabular}
|
||||
\]
|
||||
\pagebreak
|
||||
\subsection{e)}
|
||||
“All foods that are healthy to eat do not taste good.”
|
||||
“Tofu is healthy to eat.” “You only eat what tastes
|
||||
good.” “You do not eat tofu.” “Cheeseburgers are not
|
||||
healthy to eat.”
|
||||
\textbf{Assume:}\\
|
||||
P(x) = "x is healthy to eat."\\
|
||||
Q(x) = "x tastes good."\\
|
||||
R(x) = "You eat x."\\
|
||||
\[
|
||||
\begin{tabular}{lcc}
|
||||
& \textbf{Step} & \textbf{Reason} \\
|
||||
1. & \(\forall x(P(x) \implies \neg Q(x))\) & Premise. \\
|
||||
2. & P(Tofu) & Premise. \\
|
||||
3. & \(\forall x(R(x) \implies Q(x))\) & Premise. \\
|
||||
4. & \(\neg\)R(Tofu) & Premise. \\
|
||||
5. & \(\neg\)P(Cheeseburgers) & Premise. \\
|
||||
6. & P(Tofu) \(\implies\) \(\neg\)Q(Tofu) & Universal instantiation from (1). \\
|
||||
8. & \(\neg\)Q(Tofu) & Modus ponens from (2) and (6). \\
|
||||
\end{tabular}
|
||||
\]
|
||||
\pagebreak
|
||||
\problem{24}
|
||||
The error in this arguement occurs in steps 3 and 5 when the reason for the steps is stated as simplification. In order to use simplication you must have an AND statement, but in this case there is an OR statement. The correct reason for steps 3 and 5 is disjunctive syllogism.
|
||||
\end{document}
|
||||
4
hw/hw3/hw3.aux
Normal file
4
hw/hw3/hw3.aux
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{Unit 1.7}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{Unit 1.8}{7}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{10}
|
||||
159
hw/hw3/hw3.fdb_latexmk
Normal file
159
hw/hw3/hw3.fdb_latexmk
Normal file
|
|
@ -0,0 +1,159 @@
|
|||
# Fdb version 4
|
||||
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|
||||
\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{}
|
||||
|
||||
%
|
||||
% 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 3}
|
||||
\newcommand{\hmwkDueDate}{September 21, 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{1.7}
|
||||
|
||||
\problem{8}
|
||||
Prove that if \(n\) is a perfect square, then \(n + 2\) is not a perfect square.\\
|
||||
|
||||
\begin{proof}
|
||||
Let \(n\) be a perfect square. Given P(x) = \(\exists x \in \AllIntegers, n = x^2\) and Q(y) = \(\exists y \in \AllIntegers, n + 2 = y^2\), we will prove that \(\forall n \in \AllIntegers(P(n)\implies\neg Q(n))\) using a proof by contradiction.\\
|
||||
\textbf{Proof by Contradiction:}\\
|
||||
\begin{align*}
|
||||
\text{Assume } \exists n \in \AllIntegers(P(n) \implies Q(n)).&&\text{Premise.}\\
|
||||
P(x) = \exists x \in \AllIntegers, n = x^2&&\text{Premise.}\\
|
||||
Q(y) = \exists x \in \AllIntegers, n + 2 = y^2&&\text{Premise.}\\
|
||||
x^2 + 2 = y^2&&\text{Algebraic Substitution.}\\
|
||||
y^2 - x^2 = 2&&\text{Algebraic Equivalence.}\\
|
||||
(y + x)(y - x) = 2&&\text{Factorization.}\\
|
||||
\end{align*}
|
||||
Since \(y\) and \(x\) are integers, \(y + x\) and \(y - x\) are integers. The only integer factors of 2 are 1 and 2. Therefore,\\
|
||||
\begin{align*}
|
||||
y + x = 1\\
|
||||
y - x = 2
|
||||
\end{align*}
|
||||
Using algebraic equivalence, we can solve for \(y\).\\
|
||||
\begin{align*}
|
||||
y = \frac{3}{2}&&\bot\\
|
||||
\end{align*}
|
||||
Since \(y\) is not an integer, we have reached a contradiction. Therefore, \(\forall n \in \AllIntegers(P(n)\implies\neg Q(n))\).
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\problem{14}
|
||||
Prove that if \(x\) is rational and \(x \neq 0\), then \(1/x\) is rational.\\
|
||||
|
||||
\begin{proof}
|
||||
Let \(x\) be rational and \(x \neq 0\). Given \(P(x) = \exists p,q \in \AllIntegers, p,q \neq 0, x = p/q\) and \(Q(y) = \exists r,s \in \AllIntegers, r,s \neq 0, 1/x = r/s,\)k we will prove that \(\forall x \in \AllRationals(P(x)\implies Q(x))\) using a direct proof.\\
|
||||
\textbf{Direct Proof:}
|
||||
\begin{align*}
|
||||
x = \frac{p}{q}&&\text{Premise.}\\
|
||||
\frac{1}{x} = \frac{q}{p}&&\text{Algebraic Equivalence.}\\
|
||||
\frac{1}{x} = \frac{r}{s}&&\text{Let } r = q \text{ and } s = p.\\
|
||||
s \neq 0&&\text{Since } p \neq 0.\\
|
||||
\exists r,s \in \AllIntegers, r,s \neq 0, 1/x = r/s&&\text{Premise.}\\
|
||||
\forall x \in \AllRationals(P(x)\implies Q(x))&&\text{Conclusion.}
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\problem{20}
|
||||
Prove that if \(n\) is an integer and \(3n + 2\) is even, then n is even using\\
|
||||
\begin{enumerate}[a)]
|
||||
\item a proof by contraposition\\
|
||||
\textbf{Proof by Contraposition:}
|
||||
\begin{proof}
|
||||
Given \(P(n) = \exists x \in \AllIntegers, 2x = 3n + 2\) and \(Q(n) = \exists y \in \AllIntegers, 2y = n\), we will prove that \(\forall n \in \AllIntegers(P(n)\implies Q(n))\) using a proof by contraposition. In fact, we want to show that if \(n\) is odd, then \(3n + 2\) is odd.\\
|
||||
\begin{align*}
|
||||
\text{Assume } \exists n \in \AllIntegers(\neg Q(n) \implies \neg P(n)).&&\text{Premise.}\\
|
||||
\neg Q(n) = \exists y \in \AllIntegers,n = 2y + 1&&\text{Premise.}\\
|
||||
\end{align*}
|
||||
Now we need to prove \(\neg P(n)\).\\
|
||||
\begin{align*}
|
||||
3n + 2 &= 3(2y + 1) + 2&&\text{Algebraic Substitution.}\\
|
||||
&= 6y + 5 &&\text{Algebraic Equivalence.}\\
|
||||
& = 2(3y + 2) + 1&&\text{Algebraic Equivalence.}\\
|
||||
\end{align*}
|
||||
Let \(\bar{k}=3y+2, \bar{k}\in\AllIntegers.\)
|
||||
\begin{align*}
|
||||
&= 2\bar{k} + 1&&\text{Algebraic Equivalence.}\\
|
||||
\end{align*}
|
||||
Since \(\bar{k}\) is an integer, \(2\bar{k}+1\) is odd. Therefore, \(3n + 2\) is odd and we have proved the contrapositive.\\
|
||||
\end{proof}
|
||||
\pagebreak
|
||||
\item a proof by contradiction\\
|
||||
\textbf{Proof by Contradiction:}\\
|
||||
\begin{proof}
|
||||
Given \(P(n) = \exists x \in \AllIntegers, 2x = 3n + 2\) and \(Q(n) = \exists y \in \AllIntegers, 2y = n\), we will prove that \(\forall n \in \AllIntegers(P(n)\implies Q(n))\) using a proof by contradiction. In fact, we have to show a contradiction in the statement, if \(3n+2\) is even then \(n\) is odd.\\
|
||||
\begin{align*}
|
||||
\text{Assume } \exists n \in \AllIntegers(P(n) \implies \neg Q(n)).&&\text{Premise.}\\
|
||||
P(n) = \exists x \in \AllIntegers, 2x = 3n + 2&&\text{Premise.}\\
|
||||
\neg Q(n) = \exists y \in \AllIntegers, n = 2y + 1&&\text{Premise.}\\
|
||||
\end{align*}
|
||||
Now we need to find the contradiction in the statement.\\
|
||||
\begin{align*}
|
||||
2x &= 3(2y + 1) + 2&&\text{Algebraic Substitution.}\\
|
||||
&= 6y + 5 &&\text{Algebraic Equivalence.}\\
|
||||
& = 2(3y + 2) + 1&&\text{Algebraic Equivalence.}\\
|
||||
\end{align*}
|
||||
Let \(\bar{k}=3y+2, \bar{k}\in\AllIntegers.\)
|
||||
\begin{align*}
|
||||
2x &= 2\bar{k} + 1&&\bot\\
|
||||
\end{align*}
|
||||
Here we have found the contradiction in the previous statement, since \(2x\) is even and \(2\bar{k}+1\) is odd. Therefore, we have proved the proposition.\\
|
||||
\end{proof}
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\problem{30}
|
||||
Prove that \(m^2 = n^2\) if and only if \(m = n\) or \(m = -n\).
|
||||
\begin{proof}
|
||||
Given \(P(m,n) = "m^2 = n^2"\) and \(Q(m,n) = "m = n" \Or "m = -n"\), we will prove that \(\forall m,n \in \AllIntegers(P(m,n)\iff Q(m,n))\) using a direct proof.\\
|
||||
\textbf{Direct Proof:}\\
|
||||
In the case of \(P(m,n) \implies Q(m,n)\),\\
|
||||
\begin{align*}
|
||||
m^2 &= n^2\\
|
||||
m^2 - n^2 &= 0\\
|
||||
(m - n)(m + n) &= 0\\
|
||||
\end{align*}
|
||||
Which yields,
|
||||
\begin{align*}
|
||||
m - n &= 0\\
|
||||
m &= n\\
|
||||
\textbf{or}\\
|
||||
m + n &= 0\\
|
||||
m &= -n\\
|
||||
\end{align*}
|
||||
Therefore, \(P(m,n) \implies Q(m,n)\).\\\\
|
||||
In the case of \(Q(m,n) \implies P(m,n)\) we can assume that \(m = n\) without the loss of generality,\\
|
||||
\begin{align*}
|
||||
m &= n\\
|
||||
m - n &= 0\\
|
||||
(m - n)(m + n) &= 0\\
|
||||
m^2 - n^2 &= 0\\
|
||||
m^2 &= n^2\\
|
||||
\end{align*}
|
||||
Therefore, \(Q(m,n) \implies P(m,n)\).
|
||||
Since \(P(m,n) \implies Q(m,n)\) and \(Q(m,n) \implies P(m,n)\), we have proved that \(\forall m,n \in \AllIntegers(P(m,n)\iff Q(m,n))\).
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\unit{1.8}
|
||||
\problem{8}
|
||||
Prove using the notion of without loss of generality that \(5x + 5y\) is an odd integer when \(x\) and \(y\) are integers of opposite parity.
|
||||
|
||||
\begin{proof}
|
||||
Given \(P(x,y) = \exists k,l \in \AllIntegers, x = 2k, y = 2l+1\) and \(Q(x,y) = \exists w \in \AllIntegers, 2w + 1 = 5x + 5y\), we will prove that \(\forall x,y \in \AllIntegers(P(x,y)\implies Q(x,y)\) is odd) using a direct proof. Without loss of generality, we can assume that \(x\) is even and \(y\) is odd.\\
|
||||
\textbf{Direct proof:}
|
||||
\begin{align*}
|
||||
\text{Assume } &x = 2k, y = 2l + 1, k,l \in \AllIntegers.&&\text{Premise.}\\
|
||||
5x + 5y &= 5(2k) + 5(2l + 1)&&\text{Algebraic Substitution.}\\
|
||||
&= 10k + 10l + 5&&\text{Algebraic Equivalence.}\\
|
||||
&= 2(5k + 5l + 2) + 1&&\text{Algebraic Equivalence.}\\
|
||||
\end{align*}
|
||||
Let \(\bar{k} = 5k + 5l + 2, \bar{k} \in \AllIntegers\).
|
||||
\begin{align*}
|
||||
&= 2\bar{k} + 1&&\text{Algebraic Equivalence.}\\
|
||||
\end{align*}
|
||||
Therefore \(5x + 5y\) is an odd integer when \(x\) and \(y\) are integers of opposite parity.
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\problem{10}
|
||||
Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?
|
||||
|
||||
\begin{proof}
|
||||
Given \(P(n) = \exists x \in \AllIntegers, x = \sum_{i=1}^{n}i\) and \(Q(n) = \exists y \in \AllIntegers, y > 0\), we will prove that \(\forall n \in \AllIntegers(P(n)\implies Q(n))\) using a direct proof.\\
|
||||
\textbf{Constructive Proof:}
|
||||
\begin{align*}
|
||||
P(1)=\exists x \in \AllIntegers, x &= \sum_{i=1}^{1}i&&\text{Given P(1).}\\\\ &=1&&
|
||||
\end{align*}
|
||||
Using a \textbf{constructive} proof, we have shown that there exists a positive integer, 1, that equals the sum of the positive integers not exceeding it.
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\problem{32}
|
||||
Prove that there are no solutions in integers \(x\) and \(y\) to the equation \(2x^2 + 5y^2 = 14\).
|
||||
|
||||
\begin{proof}
|
||||
Given \(P(x,y) = \exists x,y \in \AllIntegers, 2x^2 + 5y^2 = 14\), we will prove that \(\forall x,y \in \AllIntegers(\neg P(x,y))\) using a proof by exhaustion.\\
|
||||
\textbf{Proof by Exhaustion:}\\
|
||||
Given \(2x^2 + 5y^2 = 14\),
|
||||
\begin{align*}
|
||||
|2x^2| &\leq 14\\
|
||||
|5y^2| &\leq 14\\
|
||||
\end{align*}
|
||||
Our possible values for \(x\) are \(\{0,1,2,3\}\) and our possible values for \(y\) are \(\{0,1,2\}\).\\
|
||||
\begin{align*}
|
||||
2(0)^2 + 5(0)^2 &= 0 + 0 = 0 &\neq 14&&(0,0)\\
|
||||
2(0)^2 + 5(1)^2 &= 0 + 5 = 5 &\neq 14&&(0,1)\\
|
||||
2(0)^2 + 5(2)^2 &= 0 + 20 = 20 &\neq 14&&(0,2)\\
|
||||
2(1)^2 + 5(0)^2 &= 2 + 0 = 2 &\neq 14&&(1,0)\\
|
||||
2(1)^2 + 5(1)^2 &= 2 + 5 = 7 &\neq 14&&(1,1)\\
|
||||
2(1)^2 + 5(2)^2 &= 2 + 20 = 22 &\neq 14&&(1,2)\\
|
||||
2(2)^2 + 5(0)^2 &= 8 + 0 = 8 &\neq 14&&(2,0)\\
|
||||
2(2)^2 + 5(1)^2 &= 8 + 5 = 13 &\neq 14&&(2,1)\\
|
||||
2(2)^2 + 5(2)^2 &= 8 + 20 = 28 &\neq 14&&(2,2)\\
|
||||
2(3)^2 + 5(0)^2 &= 18 + 0 = 18 &\neq 14&&(3,0)\\
|
||||
2(3)^2 + 5(1)^2 &= 18 + 5 = 23 &\neq 14&&(3,1)\\
|
||||
2(3)^2 + 5(2)^2 &= 18 + 20 = 38 &\neq 14&&(3,2)\\
|
||||
\end{align*}
|
||||
Therefore, \(\forall x,y \in \AllIntegers(\neg P(x,y))\).
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
\problem{36}
|
||||
Prove that \(\sqrt[3]{2}\) is irrational.
|
||||
|
||||
\begin{proof}
|
||||
Given P(x) = \(x \notin \AllRationals\), we can do a proof by contradiction to prove that \(\sqrt[3]{2}\) is irrational. Given \(\neg P(x) = \exists p,q \in \AllIntegers, q\neq 0, x = \frac{p}{q}\)\\
|
||||
\textbf{Proof by Contradiction:}\\
|
||||
\begin{align*}
|
||||
\sqrt[3]{2} &= \frac{p}{q} && \text{Given } \neg P(x)\\
|
||||
2 &= \frac{p^3}{q^3} && \text{Cubing both sides.}\\
|
||||
2q^3 &= p^3 && \text{Multiplying both sides by } q^3.\\
|
||||
\end{align*}
|
||||
Let \(\bar{k} = q^3, \bar{k} \in \AllIntegers\).
|
||||
\begin{align*}
|
||||
2\bar{k} &= p^3 && \text{Algebraic Substitution.}\\
|
||||
p^3 &\text{ is even.} && \text{Since } 2\bar{k} \text{ is even.}\\
|
||||
p &\text{ is even.} && \text{Since } p^3 \text{ is even.}\\
|
||||
\end{align*}
|
||||
Since \(p\) is even, we have two cases where \(q\) is even and \(q\) is odd.\\
|
||||
\begin{enumerate}
|
||||
\item \(q\) is even.\\
|
||||
\begin{align*}
|
||||
2|p, 2|q&&\text{Since p and q are even.}\\
|
||||
\gcd(p,q) \geq 2&&\text{Since p and q are even.}\\
|
||||
&&\bot\\
|
||||
\end{align*}
|
||||
We find the contradiction in the statement, since \(\gcd(p,q) \geq 2\) and \(p/q\) is in lowest terms.
|
||||
\item \(q\) is odd.\\
|
||||
\begin{align*}
|
||||
\exists l \in \AllIntegers, p &= 2l + 1&&\text{Since p is odd.}\\
|
||||
2q^3 &= (2l + 1)^3&&\text{Algebraic Substitution.}\\
|
||||
2q^3 &= (4l^2 + 4l + 1)(2l + 1)&&\text{Algebraic Expansion.}\\
|
||||
2q^3 &= 8l^3 + 12l^2 + 6l + 1&&\text{Algebraic Expansion.}\\
|
||||
2q^3 &= 2(4l^3 + 6l^2 + 3l) + 1&&\text{Algebraic Equivalence.}\\
|
||||
&&\bot\\
|
||||
\end{align*}
|
||||
We find the contradiction in the statement, since \(2q^3\) is even and \(2(4l^3 + 6l^2 + 3l) + 1\) is odd.
|
||||
\end{enumerate}
|
||||
Therefore, we have proved that \(\sqrt[3]{2}\) is irrational by contradiction.
|
||||
\end{proof}
|
||||
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user