Department of Mathematics and Statistics, Binghamton University people:fer:330ws:fall2018

Number Systems - Math 330 - 03 (Fall 2018)

2018-12-07T15:57:49-04:00

Number Systems - Math 330 - 03 (Fall 2018)

Fall 2018

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

Classroom:	WH-100B	MWF 11:20 - 12:50

Office Hours:	Monday	1:15 - 2:15
(subject to change)	Tuesday	4:00 - 5:00
(subject to change)	Friday	8:30 - 9:30

Announcements

Office hours during finals week:
Mon 2:30-3:30
Wed 9:30-10:30
Fri 10:00-11:00

Final Exam is Friday, Dec 14 12:50 - 2:50 in SW-112.

Homework

Appendix to Ch. 2

Appendix to Ch. 6

Homework

2018-12-07T14:11:56-04:00

Math 330 - 03 Homework (Fall 2018)

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 14 (complete) Due: 12/10/2018, optional (put inside bag on my office door)

Prove that if $A$ and $B$ are finite sets, then $A\union B$ is a finite set.
Prove that if $A$ and $B$ are countable sets, then $A\union B$ is a countable set. (Hint: use Prop. 13.9)
Prove The. 13.28 (Hint: consider the function $f(x)=\tan(x)$ from calculus)

Problem Set 13 (complete) Due: 12/03/2018. Board presentation: 12/??/2018

Prove the converse of Prop 11.2
Prove that for all $x,y,z,w\in\R$ with $z\neq 0\neq w$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
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 $a_n$.

Problem Set 12 (complete) Due: 11/19/2018. Board presentation: 11/??/2018

Prove Prop. 10.17
Prove Prop. 10.23.iii

Previous Homework

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Previous Homework

2018-11-28T15:10:45-04:00

Math 330 - 03 Homework (Fall 2018)

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 11 (complete) Due: 11/12/2018. Board presentation: 11/16/2018

Prove the following corollary to Prop. 10.4
Corollary: $\glb(\R^+)=0$.
Prove Prop. 10.7
Prove Prop. 10.10.iii
Prove Prop. 10.13.ii

Problem Set 10 (complete) Due: 11/05/2018. Board presentation: 11/14/2018

Let $f:A\to B$ and $g:B\to C$ be functions.
1. If $g\circ f$ is injective, then $f$ is injective.
2. If $g\circ f$ is surjective, then $g$ is surjective.
Construct examples of functions $f:A\to B$ and $g:B\to C$ such that:
1. $g\circ f$ is injective, but $g$ is not injective.
2. $g\circ f$ is surjective, but $f$ is not surjective.
Prove Prop. 9.15 (Hint: induction)
Prove Prop. 9.18

Problem Set 09 (complete) Due: 10/29/2018. Board presentation: 11/05/2018

Prove Prop. 8.40.ii
Prove Prop. 8.41
Prove Prop. 8.50
Give examples of subsets of $\R$ which are:
1. bounded below and above,
2. bounded below, but not bounded above,
3. bounded above, but not bounded below,
4. not bounded above or below.

Problem Set 08 (complete) Due: 10/22/2018. Board presentation: 10/31/2018

Prove Prop. 6.16
Prove Prop. 6.17
Prove Prop. 6.25 (first part)
Use Euclid's Lemma to prove the following corollary. Let $p$ be a prime, $k\in\N$, $m_1,m_2,\dots,m_k\in\N$. If $p|(m_1m_2\cdots m_k)$ then there is some $i$ with $1\leq i \leq k$ such that $p|m_i$. (Hint: Use induction on $k$).

Problem Set 07 (complete) Due: 10/15/2018. Board presentation: 10/31/2018

Let $A$ be a set, and $\sim$ an equivalence relation on $A$. Let $A/\sim$ be the partition consisting of all equivalence classes of $\sim$. Let $\Theta_{(A/\sim)}$ be the equivalence relation induced by the partition $A/\sim$. Prove that $\Theta_{(A/\sim)}=\ \sim$.
Do Project 6.8.iv.

Problem Set 06 (complete) Due: 10/08/2018. Board presentation: 10/31/2018

Prove that set union is associative.
Show, by counterexample, that set difference is not associative.
Prove Prop. 5.20.ii
Let $X$ and $Y$ be sets. Let $\power(X)$ denote the power set of $X$. Prove that: \[X\subseteq Y \iff \power(X)\subseteq\power(Y).\]
(challenge) Prove that symmetric difference is associative.

Problem Set 05 (complete) Due: 10/01/2018. Board presentation: 10/05/2018

Prove Prop. 4.6.iii
Prove Prop. 4.11.ii
Prove Prop. 4.15.i
Prove Prop. 4.16.ii

Problem Set 04 (complete) Due: 09/17/2018. Board presentation: 09/21/2018

Prove Prop. 2.38 (appendix)
Prove Prop. 2.41.iii (appendix)

Problem Set 03 (complete) Due: 09/12/2018. Board presentation: 09/17/2018

Prove that for all $k\in\N$, $k^2+k$ is divisible by 2.
Prove Prop. 2.18.iii
Prove Prop. 2.21. Hint: use proof by contradiction.
Prove Prop. 2.23. Show, by counterexample, that the statement is not true if the hypothesis $m,n\in\N$ is removed.
Fill-in the blank and prove that for all $k\geq\underline{\ \ }$, $k^2 < 2^k$.

Problem Set 02 (complete) Due:09/05/2018. Board presentation: 09/10/2018

Prove Prop. 1.24
Prove Prop. 1.27.ii,iv
Prove Prop. 2.7.i,ii
Prove Prop. 2.12.iii

Problem Set 01 (complete) Due: 08/27/2018. Board presentation: 08/31/2018

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

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