diff --git a/hw/hw3/hw3.fdb_latexmk b/hw/hw3/hw3.fdb_latexmk index 2b3c5de..dd0ee9d 100644 --- a/hw/hw3/hw3.fdb_latexmk +++ b/hw/hw3/hw3.fdb_latexmk @@ -1,7 +1,7 @@ # Fdb version 4 -["pdflatex"] 1695163984 "/home/rushil/school/math/hw/hw3/hw3.tex" "hw3.pdf" "hw3" 1695163985 0 +["pdflatex"] 1695319522 "/home/rushil/school/math/hw/hw3/hw3.tex" "hw3.pdf" "hw3" 1695319523 0 "/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 "" - "/home/rushil/school/math/hw/hw3/hw3.tex" 1695163984 14978 a4f3ff2fb1cae2a1d04c663ddcc8ed79 "" + "/home/rushil/school/math/hw/hw3/hw3.tex" 1695319522 14989 87bc0b0b38513df51e06cb1a48f3c28e "" "/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 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -150,8 +150,8 @@ "/usr/share/texmf/web2c/texmf.cnf" 1681034085 39561 34c98e380bf7c7201ee6a7909aff625a "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1693368379 4623455 fa0568a71dd9a288d6c226ee477506c6 "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1693368408 7880218 5644a8bb2704ddc1060e0c139ce5389a "" - "hw3.aux" 1695163985 195 4c8818449daf66d8271d10ab24c41f38 "pdflatex" - "hw3.tex" 1695163984 14978 a4f3ff2fb1cae2a1d04c663ddcc8ed79 "" + "hw3.aux" 1695319523 195 4c8818449daf66d8271d10ab24c41f38 "pdflatex" + "hw3.tex" 1695319522 14989 87bc0b0b38513df51e06cb1a48f3c28e "" (generated) "hw3.aux" "hw3.log" diff --git a/hw/hw3/hw3.log b/hw/hw3/hw3.log index fd5542f..b9cec8c 100644 --- a/hw/hw3/hw3.log +++ b/hw/hw3/hw3.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 19 SEP 2023 18:53 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 21 SEP 2023 14:05 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -555,22 +555,22 @@ Underfull \hbox (badness 10000) in paragraph at lines 324--326 [] [9] -Underfull \hbox (badness 10000) in paragraph at lines 348--350 +Underfull \hbox (badness 10000) in paragraph at lines 343--345 [] -Underfull \hbox (badness 10000) in paragraph at lines 360--363 +Underfull \hbox (badness 10000) in paragraph at lines 357--360 [] -Underfull \hbox (badness 10000) in paragraph at lines 363--364 +Underfull \hbox (badness 10000) in paragraph at lines 360--361 [] -Underfull \hbox (badness 10000) in paragraph at lines 370--371 +Underfull \hbox (badness 10000) in paragraph at lines 367--368 [] @@ -578,13 +578,13 @@ Underfull \hbox (badness 10000) in paragraph at lines 370--371 Here is how much of TeX's memory you used: 15643 strings out of 476091 318305 string characters out of 5794081 - 1871330 words of memory out of 5000000 + 1876330 words of memory out of 5000000 35750 multiletter control sequences out of 15000+600000 522421 words of font info for 70 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 102i,12n,104p,445b,374s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on hw3.pdf (10 pages, 171963 bytes). +Output written on hw3.pdf (10 pages, 172123 bytes). PDF statistics: 110 PDF objects out of 1000 (max. 8388607) 69 compressed objects within 1 object stream diff --git a/hw/hw3/hw3.pdf b/hw/hw3/hw3.pdf index 756c981..b5c4849 100644 Binary files a/hw/hw3/hw3.pdf and b/hw/hw3/hw3.pdf differ diff --git a/hw/hw3/hw3.synctex.gz b/hw/hw3/hw3.synctex.gz index 4b617fe..ec50976 100644 Binary files a/hw/hw3/hw3.synctex.gz and b/hw/hw3/hw3.synctex.gz differ diff --git a/hw/hw3/hw3.tex b/hw/hw3/hw3.tex index 4716f3f..5fcb3a6 100644 --- a/hw/hw3/hw3.tex +++ b/hw/hw3/hw3.tex @@ -322,20 +322,15 @@ Prove that there are no solutions in integers \(x\) and \(y\) to the equation \( |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 \(\{0,1,2\}\).\\ + Our possible values for \(x\) are \(\{0,1,2\}\) and our possible values for \(y\) are \(\{0,1\}\).\\ \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} @@ -351,32 +346,37 @@ Prove that \(\sqrt[3]{2}\) is irrational. \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*} + Given that p is even, Let \(\bar{k} \in \AllIntegers, p=2\bar{k}\). + \begin{align*} + 2q^3&= (2\bar{k})^3 && \text{Algebraic Substitution.}\\ + 2q^3&= 8\bar{k}^3 && \text{Algebraic Equivalence.}\\ + q^3&= 4\bar{k}^3 && \text{Algebraic Equivalence.}\\ + \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.}\\ + \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.}\\ + \exists l \in \AllIntegers, q &= 2l + 1&&\text{Since q is odd.}\\ + (2l + 1)^3 &= 4\bar{k}^3&&\text{Algebraic Substitution.}\\ + 8l^3 + 12l^2 + 6l + 1 &= 4\bar{k}^3&&\text{Algebraic Expansion.}\\ + 8l^3 +12l^2 +6l-4\bar{k}^3 &= -1&&\text{Algebraic Equivalence.}\\ + 2(4l^3 + 6l^2 + 3l-2\bar{k}^3) &= -1&&\text{Algebraic Equivalence.}\\ + \bar{l} \in \AllIntegers, \bar{l} &= 4l^3 + 6l^2 + 3l-2\bar{k}^3&&\text{Define } \bar{l}\\ + 2\bar{l} &= -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. + + We find the contradiction in the statement, since \(2\bar{l}\) cannot equal \(-1\). \end{enumerate} Therefore, we have proved that \(\sqrt[3]{2}\) is irrational by contradiction. \end{proof} diff --git a/notes/notes.aux b/notes/notes.aux new file mode 100644 index 0000000..a1ae030 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+\usepackage{tikz} +\usepackage[plain]{algorithm} +\usepackage{algpseudocode} +\usepackage[shortlabels]{enumitem} +\usepackage{mathtools} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{babel} + + + +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ (\hmwkClassInstructor\ \hmwkClassTime): \hmwkTitle} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} + +% +% 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}{Notes} +\newcommand{\hmwkDueDate}{Fall, 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}} +\newcommand{\AllNaturals}{\mathbb{N}} + +\newtheorem{proposition}{Proposition} +\newtheorem{theorem}{Theorem} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{proposition} + The sum of the first n, positive odd integers is the equation \(n^2\). +\end{proposition} + +\begin{proof} + By induction + \begin{enumerate} + \item base case + \begin{align*} + P(1) &= 1^2 = 1\\ + \end{align*} + \item inductive step + \begin{align*} + \text{Assume } P(n) = "1+3+5+...+2n-1 = n^2"\\ + \text{WTS: } P(n+1) = "1+3+5+...+2n-1+2(n+1)-1 = (n+1)^2"\\ + 1+3+5+...+2n-1+2(n+1)-1 &= (n+1)^2\\ + \end{align*} + \end{enumerate} +\end{proof} +\begin{proposition} All horses are the same color.\\ +\emph{Note:} It suffices to show: P(n) = "All sets of n horses have the same color"\\ +\begin{proof} + By induction + \begin{enumerate} + \item base case:\\ + P(1) = "All sets of 1 horse have the same color"\\ + \item inductive step: Assume P(n), i.e. every step of n horses have the same color\\ + WTS: P(n+1), i.e. every set of n+1 horses have the same color\\ + H = \(\{H_1, H_2, ..., H_n,H_{n+1}\}\) is a set of n+1 horses\\ + \(H_1 = \{H_1, H_2, ..., H_n\}\) is a set of n horses\\ + \(H_2 = \{H_2, H_3, ..., H_n,H_{n+1}\}\) is a set of n horses\\ + \end{enumerate} +\end{proof} +\end{proposition} + +\begin{theorem} + Given two sets, when \(n \neq 1\), when they both overlap and are disjoint, the union of the two sets is equal to the sum of the two sets. +\end{theorem} +\pagebreak +\section{Strong Induction} +This is what we were doing before:\\ +\textbf{Weak Induction:} +\begin{enumerate} + \item base case + \item inductive step (\(P(n) \implies P(n+1)\)) +\end{enumerate} +Trying to prove \(P(n) \forall n \in \AllNaturals\).\\\\ +If \(P(n) \implies P(n+1)\) is too hard to show, instead try strong induction: +\begin{enumerate} + \item base case (Assume all steps before \(n+1\)) + \item Assume \(P(k)\forall k \in \{1,2,...,n\}\) then try to show P(n+1) +\end{enumerate} +\begin{proposition} + A chocolate bar with n \(\geq\) 1 pieces can be broken into individual pieces by making n-1 breaks. +\end{proposition} +\begin{proof} + Using weak induction, + \begin{enumerate} + \item base case: n = 1\\ + 1 piece can be broken into individual pieces by making 0 breaks. + \item inductive step\\ + Assume P(n)="a bar with n pieces can be broken into individual pieces by making n-1 breaks"\\ + WTS: P(n+1)="a bar with n+1 pieces can be broken into individual pieces by making n breaks"\\ + The issue is that we need to know that everything from P(n) to P(1) works.\\ + \textbf{Since we cannot prove this for an arbitrary n, we must use strong induction.}\\ + \end{enumerate} +\end{proof} +\begin{proof} + Using strong induction, + \begin{enumerate} + \item base case: n = 1\\ + 1 piece can be broken into individual pieces by making 0 breaks. + \item inductive step\\ + Assume P(k) \(\forall k \in \{1,2,...,n\}\)\\ + WTS: P(n+1)="a bar with n+1 pieces can be broken into individual pieces by making n breaks"\\ + \begin{enumerate} + \item Consider an arbitrary bar of n+1 size. Break the bar into two pieces + \begin{enumerate} + \item One piece has k pieces + \item The other piece has \((n+1)-k\) pieces + \end{enumerate} + \item Assuming \(P(k)\), the first piece can be broken into individual pieces by making \(k-1\) breaks. + \item \(P(n+1-k)\) will require \(n+1-k-1=n-k\) breaks. + \end{enumerate} + \(\therefore\) The total number of breaks is \(1+(k-1)+(n-k)=n\). + \end{enumerate} +\end{proof} +\pagebreak +\begin{theorem} + It is true that strong induction \(\rightarrow\) weak induction +\end{theorem} +\begin{theorem} + Fundamental Theorem of Arithmetic: + \begin{align*} + \forall n \in \AllNaturals - {0,1}, \text{n is either prime or can be written as a product of primes.}\\ + \end{align*} +\end{theorem} +\begin{proof} + Using strong induction, + \begin{enumerate} + \item base case: n = 2\\ + 2 is prime. + \item inductive step: Assume P(k) \(\forall k \in \{2,3,...,n\}\)\\ + WTS: P(n+1)\\ + We can prove this by cases: + \begin{enumerate} + \item n+1 is prime: + It can be expressed as \(1 \times (n+1)\) + \item n+1 is not prime\\ + \begin{align*} + \exists l,w \in \AllIntegers, n+1 = lw\\ + 1 < l, w < n+1\\ + \end{align*} + So P(l) = T, P(w) = T, therefore:\\ + \(l = p_1^{k_1}\times p_2^{k_2} \times ... \times p_n^{k_n}\), where \(p_1, p_2, ..., p_n\) are primes and \(k\in \AllNaturals\).\\ + \(w = q_1^{j_1}\times q_2^{j_2} \times ... \times q_m^{j_m}\), where \(q_1, q_2, ..., q_m\) are primes and \(j\in \AllNaturals\).\\\\ + Given \(l\) and \(w\), we can find \(n+1\) by multiplying them together.\\ + \begin{align*} + n+1 &= lw\\ + n+1 &= (p_1^{k_1}\times p_2^{k_2} \times ... \times p_n^{k_n}) (q_1^{j_1}\times q_2^{j_2} \times ... \times q_m^{j_m})\\ + \end{align*} + \end{enumerate} + \end{enumerate} + \(n+1\) is a product of primes.\\ +\end{proof} +\pagebreak +\begin{proposition} + Consider the sequence \(a_1=0, a_2=1, a_n=2a_{n-1}-a_{n-2}\). Prove \(a_n = n-1\). +\end{proposition} +\begin{proof} + Using strong induction, + \begin{enumerate} + \item bsae case: n = 1, n=2\\ + \(n=1: a_1 = 0 = 1-1\)\\ + \end{enumerate} +\end{proof} +\end{document} \ No newline at end of file