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Math 330 - 01 Homework (Spring 2022)

  • LaTeX-ed solutions are encouraged and appreciated.
  • If you use LaTeX, hand-in a printed version of your homework.
  • You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange of any written material.
  • Writing of homework problems should be done on an individual basis.
  • References to results from the textbook and/or class notes should be included.
  • The following lists should be considered partial and tentative lists until the word complete appears next to it.
  • Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.

$\newcommand{\aut}{\textrm{Aut}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\join}{\vee} \newcommand{\bigjoin}{\bigvee} \newcommand{\meet}{\wedge} \newcommand{\bigmeet}{\bigwedge} \newcommand{\normaleq}{\unlhd} \newcommand{\normal}{\lhd} \newcommand{\union}{\cup} \newcommand{\intersection}{\cap} \newcommand{\bigunion}{\bigcup} \newcommand{\bigintersection}{\bigcap} \newcommand{\sq}[2][\ ]{\sqrt[#1]{#2\,}} \newcommand{\pbr}[1]{\langle #1\rangle} \newcommand{\ds}{\displaystyle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\mathbb{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\imp}{\Rightarrow} \newcommand{\rimp}{\Leftarrow} \newcommand{\pinfty}{1/p^\infty} \newcommand{\power}{\mathcal{P}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calT}{\mathcal{T}} \newcommand{\gal}{\textrm{Gal}} \newcommand{\isom}{\approx} \newcommand{\glb}{\textrm{glb}} $

Problem Set 13 (complete) Due: 05/09/2022

  1. Let $f:A\to B$ and $g:C\to D$ be functions. Define $f\times g:A\times C \to B\times D$ by $(f\times g)(a,c)=(f(a),g(c))$.
    Prove that if $f$ and $g$ are surjective, then so is $f\times g$.
  2. Prove that the function $\ f:\Z \to \N$ given by \[ f(m) = \cases {2m &if $m>0,$ \cr -2m+1 &if $m\leq 0,$ \cr} \] is bijective.
  3. Prove that if $A$ and $B$ are finite sets, then so is $A\union B$. Morevoer, if $A$ and $B$ are disjoint, then $|A\union B|=|A|+|B|$.
  4. Prove Theorem 13.28. Hint: consider the function $\tan(x)$ from calculus.

Problem Set 12 (complete) Due: 05/02/2022. Board presentation: 05/06/2022

  1. Prove the converse of Prop 11.2
  2. Prove that for all $x,y,z,w\in\R$ with $z\neq 0\neq w$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
  3. Consider the set $$A=\{x\in\Q\mid x^2<2\}$$ Show that $A$ is non-empty and has an upper bound in $\Q$, but does not have a least upper bound in $\Q$. Hint: by way of contradiction, assume $A$ has a least upper bound $u$ in $\Q$, and compare it with $\sqrt{2}$.
  4. Prove Prop. 11.21.iii

Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022

  1. Prove part (iv) of lemma stated in class:
    for $x\in\R$ and $r\in\R^+$,
    (iv) $|x| \leq r$ iff $x \leq r$ and $-x \leq r$.
    (Hint: use part (iii) of the same lemma.
  2. Prove Prop. 10.10.iii (Hint: use 10.8.iv)
  3. Prove Prop. 10.13.ii
  4. Prove Prop. 10.17 (Hint: use induction)

Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022

  1. Let $f:A\to B$ and $g:B\to C$ be functions.
    1. Prove Prop. 9.7.ii
    2. Prove that if $g\circ f$ is surjective, then $g$ is surjective.
  2. Prove Prop. 9.10.ii
  3. Prove Prop. 9.15 (Hint: induction)
  4. Prove Prop. 9.18

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people/fer/330ws/spring2022/homework.txt · Last modified: 2022/05/06 06:51 by fer