people:fer:330ws:fall2018:homework

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.

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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

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