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

1. Prove that if $A$ and $B$ are finite sets, then $A\union B$ is a finite set.
2. Prove that if $A$ and $B$ are countable sets, then $A\union B$ is a countable set. (Hint: use Prop. 13.9)
3. 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

1. Prove the converse of Prop 11.2
2. Prove that for all $x,y,z,w\in\R$ with $z\neq 0\neq w$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
3. 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 $\sqrt{2}$.
4. 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

1. Prove Prop. 10.17
2. Prove Prop. 10.23.iii