Processing math: 0%

User Tools

Site Tools


people:fer:330ws:fall2017:old_homework

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} \newcommand{\glb}{\textrm{glb}}

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

Problem Set 7 (complete) Due: 10/27/2017. Board presentation: 10/27/2017

  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
  3. Prove Prop. 8.40.ii
  4. Prove Prop. 8.41

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

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

Current Homework

Home

people/fer/330ws/fall2017/old_homework.txt · Last modified: 2018/08/21 11:01 by fer