Department of Mathematics and Statistics, Binghamton University people:fer:330ws:spring2022

Number Systems - Math 330 - 01 (Spring 2022)

2022-04-24T08:46:10-04:00

Number Systems - Math 330 - 01 (Spring 2022)

Spring 2022

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

Classroom:	WH-100B	MWF 8:00 - 9:30

Office Hours:	Monday	10:00 - 11:00
(subject to change)	Tuesday	4:00 - 5:00
(subject to change)	Friday	10:00 - 11:00

Announcements

The third test will be on Monday 04/25/2022.

Homework

Appendix to Ch. 2

Appendix to Ch. 6

Homework

2022-05-06T06:51:46-04:00

Math 330 - 01 Homework (Spring 2022)

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 13 (complete) Due: 05/09/2022

Let $f:A\to B$ and $g:C\to D$ be functions. Define $f\times g:A\times C \to B\times D$ by $(f\times g)(a,c)=(f(a),g(c))$.
Prove that if $f$ and $g$ are surjective, then so is $f\times g$.
Prove that the function $\ f:\Z \to \N$ given by \[ f(m) = \cases {2m &if $m>0,$ \cr -2m+1 &if $m\leq 0,$ \cr} \] is bijective.
Prove that if $A$ and $B$ are finite sets, then so is $A\union B$. Morevoer, if $A$ and $B$ are disjoint, then $|A\union B|=|A|+|B|$.
Prove Theorem 13.28. Hint: consider the function $\tan(x)$ from calculus.

Problem Set 12 (complete) Due: 05/02/2022. Board presentation: 05/06/2022

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 $\sqrt{2}$.
Prove Prop. 11.21.iii

Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022

Prove part (iv) of lemma stated in class:
for $x\in\R$ and $r\in\R^+$,
(iv) $|x| \leq r$ iff $x \leq r$ and $-x \leq r$.
(Hint: use part (iii) of the same lemma.
Prove Prop. 10.10.iii (Hint: use 10.8.iv)
Prove Prop. 10.13.ii
Prove Prop. 10.17 (Hint: use induction)

Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022

Let $f:A\to B$ and $g:B\to C$ be functions.
1. Prove Prop. 9.7.ii
2. Prove that if $g\circ f$ is surjective, then $g$ is surjective.
Prove Prop. 9.10.ii
Prove Prop. 9.15 (Hint: induction)
Prove Prop. 9.18

Previous Homework

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Previous Homework

2022-04-21T19:59:16-04:00

Math 330 - 01 Homework (Spring 2022)

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.

Previous Homework

Problem Set 09 (complete) Due: 04/04/2022. Board presentation: 04/08/2022

Prove Prop. 8.40.ii
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.
Project 9.3

Problem Set 08 (complete) Due: 03/28/2022. Board presentation: 04/01/2022

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 $k$).
Prove Prop. 6.28

Problem Set 07 (complete) Due: 03/21/2022. Board presentation: 03/25-28/2022

Prove that set union is associative.
Prove Prop. 5.20.ii
Let $X$ and $Y$ be sets. Let $\power(X)$ denote the power set of $X$. Prove that: \[X\subseteq Y \iff \power(X)\subseteq\power(Y).\]
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 the two equivalence relations $\Theta_{(A/{\sim})}$ and $\sim$ are equal.

Problem Set 06 (complete) Due: 03/07/2022. Board presentation: 03/11/2022

Prove that $$\sum_{k=2}^n \binom{k}{2} = \binom{n+1}{3}$$ Hint: use induction on $n$.
Prove that for $k\geq 1$, $$\sum_{m=0}^k (-1)^m \binom{k}{m} = 0$$
Determine the base case, and prove by induction and using the recursive definition of Fibonacci numbers that $$f_{2k}=f_{k+1}^2-f_{k-1}^2$$
Prove Prop. 4.31 without using Prop. 4.29

Problem Set 05 (complete) Due: 02/28/2022. Board presentation: 03/04/2022

Prove Prop. 4.6.iii
Prove Prop. 4.11.ii
Prove Prop. 4.15.i
Prove Prop. 4.18

Problem Set 04 (complete) Due: 02/21/2022. Board presentation: 02/28/2022

Prove Prop. 2.38 (appendix)
Prove Prop. 2.41.iii (appendix)
Project 3.1

Problem Set 03 (complete) Due: 02/14/2022. Board presentation: 02/18/2022

Prove that for all $k\in\N$, $k^2+k$ is divisible by 2. Is this true for all $k\in\Z$?
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:02/07/2022. Board presentation: 02/11/2022

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: 01/31/2022. Board presentation: 02/04/2022 (rescheduled for 02/07/2022)

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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