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Math 330 - 02 Homework

  • 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} $

Problem Set 10 (complete) Due: 11/17/2017. Board Presentation: 11/17/2017

  1. Prove Prop. 10.23.v
  2. Prove The. 10.26
  3. Let $(a_n)$ be a sequence. Consider the sequence of even-indexed terms, $(a_{2n})$, and the sequence of odd-indexed terms, $(a_{2n+1})$. Prove that if both $(a_{2n})$ and $(a_{2n+1})$ converge to $L$, then $(a_n)$ converges to $L$.
  4. Let $q_n=\displaystyle\frac{f_n}{f_{n+1}}$, where $f_n$ is the $n$-th Fibonacci number. Show that the sequence $(q_n)$ converges. What value does it converge to?

Problem Set 9 (complete) Due: 11/10/2017. Board presentation: 11/10/2017

  1. Prove Prop. 10.10.iii
  2. Prove Prop. 10.17
  3. Prove Prop. 10.23.iii

Problem Set 8 (complete) Due: 11/03/2017. Board presentation: 11/03/2017

  1. Prove Prop. 8.50
  2. Prove that function composition is associative, when defined.
  3. Let $A,B,C$ be sets and $f:A\to B$ and $g:B\to C$ functions. Prove that if $g\circ f$ is surjective then $g$ is surjective. Give an example when $g\circ f$ is surjective, but $f$ is not.
  4. Construct an example of a function with several right inverses.
  5. Prove Prop. 9.15 (Hint: induction on $k$)
  6. Prove Prop. 9.18

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people/fer/330ws/330ws_homework.txt · Last modified: 2017/11/15 12:36 by fer