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Math 330 - 02 Previous 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 5 (complete) Due: 10/06/2017. Board presentation: 10/18/2017

  1. Let $n\in\N$. Prove that if $n$ is divisible by 3, then $f_n$ is even. Is the converse true? If so, prove it; if not, give a counterexample.
  2. Let $n\in\N$. Prove that if $n$ is divisible by 5, then $f_n$ is divisible by 5. Is the converse true? If so, prove it; if not, give a counterexample.
  3. Prove the following identities for the Fibonacci numbers \[ f_{2n+1}=f_n^2+f_{n+1}^2;\quad \\ f_{2n}=f_{n+1}^2-f_{n-1}^2 = f_n(f_{n+1}+f_{n-1}) \]
  4. Prove the associativity of the set union and set intersection operations. Give a counterexample to show that set difference is not associative.

Problem Set 4 (complete) Due: 09/29/2017. Board presentation: 10/06/2017

  1. Prove Prop. 4.6.iii
  2. Prove Prop. 4.11.ii
  3. Do project 4.12
  4. Prove Prop. 4.16.ii

Problem Set 3 (complete) Due: 09/15/2017. Board presentation: 09/20/2017

  1. Prove Prop. 2.21 (Hint: proof by contradiction)
  2. Prove Prop. 2.23. Show, by counterexample, that the statement is not true when the hypothesis $m,n\in\N$ is removed.
  3. Prove Prop. 2.38 (appendix)
  4. Prove Prop. 2.41.iii (appendix)

Problem Set 2 (complete) Due: 09/08/2017. Board presentation: 09/15/2017

  1. Prove Prop. 1.25
  2. Prove Prop. 1.27.iv
  3. Prove Prop. 2.7
  4. Prove transitivity of $"\leq"$.

Problem Set 1 (complete) Due: 09/01/2017. Board Presentation: 09/08/2017

  1. Prove Prop. 1.7
  2. Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction). Can you prove that 1 + 1 ≠ 0?
  3. Prove Prop. 1.11.iv
  4. Prove Prop. 1.14

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people/fer/330ws/330ws_previous_homework.txt · Last modified: 2017/10/20 12:52 by fer