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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 7 (partial) Due: 10/27/2017. Board presentation: 11/??/2017

  1. Prove the corollary to Prop. 6.25: Let $a,b\in\Z$, $n\in\N$ and $k\geq 0$. If $a \equiv b \pmod{n}$ then $a^k \equiv b^k \pmod{n}$. (Hint: induction on $k$)
  2. Prove Prop. 8.6

Problem Set 6 (complete) Due: 10/13/2017. Board presentation: 10/20/2017

  1. Let $f_n$ be the $n$-th Fibonacci number. Prove by induction on $n$ that \[ \sum_{j=1}^n f_{2j} = f_{2n+1}-1 \]
  2. Find and write down all the partitions on a 4-element set $A=\{a,b,c,d\}$. How many equivalence relations are there on $A$?
  3. Prove Prop. 6.15
  4. Prove Prop. 6.16

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people/fer/330ws/330ws_homework.txt · Last modified: 2017/10/20 12:53 by fer