Department of Mathematics and Statistics, Binghamton University people:fer:330ws:fall2017

Number Systems - Math 330 - 02 (Fall 2017)

2018-08-24T09:02:49-04:00

Number Systems - Math 330 - 02 (Fall 2017)

Fall 2017

Instructor:	Fernando Guzmán	WH-116	x-72876	fer@math.binghamton.edu

Classroom:	WH-100B	MWF 9:40 - 11:10

Office Hours:	Tuesday	12:00 - 1:00
(subject to change)	Wednesday	3:00 - 4:00
(subject to change)	Friday	12:00 - 1:00

Announcements

Office hours for finals week:
- Monday 12/11, 3:00-4:00
- Wednesday 12/13, 3:00-4:00
- Thursday 12/14, 11:00-12:00
Tuesday office hour changes to 12:00-1:00 for the rest of the semester
Office hour for Tuesday, Nov. 7, is moved to 12:00-1:00 pm
Office hour for Wednesday, October 11, is moved to 2:00-3:00 pm
If you are interested in joining the Math Club, please send an e-mail to the Club's President, Alex Thornton athorn1@binghamton.edu

Homework

Appendix to Ch. 2

Appendix to Ch. 6

Homework

2018-08-24T09:05:50-04:00

Math 330 - 02 Homework (Fall 2017)

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 12 (complete) Due: 12/08/2017. Board presentation: 12/08/2017

Prove that if $A$ and $B$ are finite sets, then $A\union B$ is a finite set.
Prove the following corollary to Proposition 13.6.
1. If $f:A\to B$ is injective and $B$ is finite, then $A$ is finite.
2. If $g:A\to B$ is surjective and $A$ is finite, then $B$ is finite.
Do Project 13.15, finding a formula for the bijection in the picture.
Prove Theorem 13.28.

Problem Set 11 (complete) Due: 12/01/2017. Board Presentation: 12/01/2017

Write down the details of the proofs that the sum of a rational number and an irrational number is irrational, and that the product of a non-zero rational number and an irrational number is irrational.
Prove the converse of Prop. 11.2
Do Project 11.14
Prove that for all $x,y,z,w\in\R$ with $z,w\neq 0$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
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 $\sqrt{2}$.
Consider the sequence defined recursively by $$a_n=a_{n-1}+3a_{n-2} \\ a_1=1 \\ a_2=2.$$ Use the converse of Proposition 11.25 to find a closed formula for $a_n$.

Previous Homework

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Previous Homework

2018-08-21T11:01:30-04:00

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.

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

Prove Prop. 10.23.v
Prove The. 10.26
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 $L$.
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

Prove Prop. 10.10.iii
Prove Prop. 10.17
Prove Prop. 10.23.iii

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

Prove Prop. 8.50
Prove that function composition is associative, when defined.
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.
Construct an example of a function with several right inverses.
Prove Prop. 9.15 (Hint: induction on $k$)
Prove Prop. 9.18

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

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 $k$)
Prove Prop. 8.6
Prove Prop. 8.40.ii
Prove Prop. 8.41

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

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

Problem Set 5 (complete) Due: 10/06/2017. Board presentation: 10/18/2017

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.
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.
Prove the following identities for the Fibonacci numbers \[ f_{2n+1}=f_n^2+f_{n+1}^2;\quad \\ f_{2n}=f_{n+1}^2-f_{n-1}^2 = f_n(f_{n+1}+f_{n-1}) \]
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

Prove Prop. 4.6.iii
Prove Prop. 4.11.ii
Do project 4.12
Prove Prop. 4.16.ii

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

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

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

Prove Prop. 1.25
Prove Prop. 1.27.iv
Prove Prop. 2.7
Prove transitivity of $"\leq"$.

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

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

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