\documentclass[11pt]{article} \usepackage{fullpage} \usepackage{amssymb,amsfonts,amsmath,amsthm} \usepackage[margin= 1in]{geometry} \usepackage{url} \usepackage{enumerate} \usepackage{color} \usepackage[pdftex]{graphicx} \usepackage{lastpage} \usepackage{mathptmx} \newtheorem{thm}{THEOREM}[] \newtheorem{lem}[thm]{LEMMA} \newtheorem{cor}[thm]{COROLLARY} \newtheorem{prop}[thm]{PROPOSITION} \newtheorem{as}[thm]{ASSUMPTION} \newtheorem{conjecture}[thm]{Conjecture} \newtheorem{claim}[thm]{CLAIM} \theoremstyle{definition} \newtheorem{defi}[]{DEFINITION} \theoremstyle{definition} \newtheorem{remark}[thm]{REMARK} \newtheorem{fact}[thm]{Fact} \newtheorem{ex}[thm]{Example} \setlength{\parindent}{0cm} \newcommand{\tab}{\hspace{3ex}} \newcommand{\R}{\mathbb{R}} \usepackage{titlesec} \usepackage{sectsty} \hyphenchar\font=-1 \sectionfont{ \normalsize \sectionrule{0pt}{0pt}{-5pt}{0.05pt} } \titleformat*{\section}{ \Large \center} \titleformat{\subsection}[runin] {\normalfont\bfseries } {}{0em}{}[.] \titlespacing{\section} {\parindent}{0pt}{5pt} \titlespacing{\subsection} {0pt}{2ex plus .1ex minus .2ex}{5pt} \makeatletter \renewcommand{\@evenfoot}{% Page \thepage{} of \pageref{VeryLastPage} \hfil } \renewcommand{\@oddfoot}{\@evenfoot} \makeatother \begin{document} \begin{center} {\huge \sc Homework 1} \\ \smallskip { Due Date: 04/05/2022}\\ \smallskip {Name: }\\ {\large } \end{center} \vspace{1cm} \begin{enumerate} \item[1.1)] Let $A$ be the set $\{\alpha,\{1,\alpha \},\{3\}, \{ \{1,3 \}\},3 \}$. Which of the following statements are true and which are false? \begin{enumerate} \item $\alpha \in A$. \item $\{\alpha \}\notin A.$ \item $\{1,\alpha \}\subseteq A. $ \item $\{3,\{3\} \}\subseteq A. $ \item $\{1,3 \}\in A$. \item $\{\{1,3 \}\}\subseteq A$. \item $\{\{1,\alpha \}\}\subseteq A$. \item $\{1,\alpha \}\notin A$. \item $\emptyset \subseteq A$. \end{enumerate} \newpage \item[1.2)] Let $B,C,D,E$ be the following sets: \begin{equation} \label{eq1} \begin{split} B & = \{x \ |\ x \text{ is a real number}, x^2<4 \}, \\ C & = \{x \ |\ x \text{ is a real number}, 0\leq x < 2 \}, \\ D & = \{x \ |\ x\in \mathbb{Z}, x^2<1 \},\\ E& = \{1\}. \end{split} \end{equation} \begin{enumerate} \item Which pair of these sets has the property that neither is contained in the other? \item You are given that $X$ is one of the sets $B,C,D,E$, but you do not know which one. You are also given that $E\subset X$ and $X\subset B$. What can you deduce about $X$? \end{enumerate} \newpage \item[1.3)] Which of the following arguments are valid? For the valid ones, write down the argument symbolically. \begin{enumerate} \item I eat chocolate if I am depressed. I am not depressed. Therefore I am not eating chocolate. \item I eat chocolate only if I am depressed. I am not depressed. Therefore I am not eating chocolate. \item If a movie is not worth seeing, then it was not made in England. A movie is worth seeing only if critic Ivor Smallbrain reviews it. The movie Cat on a Hot Tin Proof was not reviewed by Ivor Smallbrain. Therefore Cat on a Hot Tin Proof was not made in England. \end{enumerate} \newpage \item[1.5)] Which of the following statements are true, and which are false? \begin{enumerate} \item $n=3$ only if $n^{2}-2 n-3=0$ \item $n^{2}-2 n-3=0$ only if $n=3$ \item If $n^{2}-2 n-3=0$ then $n=3$ \item For integers $a$ and $b, a b$ is a square only if both $a$ and $b$ are squares. \item For integers $a$ and $b, a b$ is a square if both $a$ and $b$ are squares. \end{enumerate} \newpage \item[1.6)]Write down careful proofs of the following statements: \begin{enumerate} \item $\sqrt{6}-\sqrt{2}>1$ \item If $n$ is an integer such that $n^{2}$ is even, then $n$ is even. \item If $n=m^{3}-m$ for some integer $m$, then $n$ is a multiple of 6 \end{enumerate} \newpage \item[1.7)] \begin{enumerate} \item If $n$ and $k$ are positive integers, then $n^k-n$ is always divisible by $k$. \item Every positive integer is the sum of three squares. \end{enumerate} \newpage \item[1.9)] In this question I am assuming you know what a prime number is; if not, take a look at the definition on page 69. For each of the following statements, form its negation and either prove that the statement is true or prove that its negation is true: \begin{enumerate} \item $\forall n\in\mathbb{Z}$ such that $n$ is a prime number, $n$ is odd. \item $\forall n\in \mathbb{Z}, \exists a,b,c,d,e,f,g,h\in \mathbb{Z}$ such that \[ n=a^3+b^3c^3+d^3+e^3+f^3+g^3+h^3. \] \item $\exists x\in\mathbb{Z}$ such that $\forall n\in \mathbb{Z}$, $x\neq n^2+2$. \item $\exists x\in\mathbb{Z}$ such that $\forall n\in \mathbb{Z}$, $x\neq n+2$. \item $\forall y\in \{x\ |\ x\in \mathbb{Z}, x\geq 1 \}, 5y^2+5y+1 $ is a prime number. \item $\forall y\in \{x\ |\ x\in \mathbb{Z}, x^2< 0 \}, 5y^2+5y+1 $ is a prime number. \end{enumerate} \end{enumerate} \newpage \begin{enumerate} \item [17.3)] Work out $\bigcup_{n=1}^{\infty} A_{n}$ and $\bigcap_{n=1}^{\infty} A_{n}$, where $A_{n}$ is defined as follows for $n \in \mathbb{N}$ : \begin{enumerate} \item $A_{n}=\{x \in \mathbb{R} \mid x>n\}$. \item $A_{n}=\left\{x \in \mathbb{R} \mid \frac{1}{n}