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## Math 330 - 02 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 12 (complete) Due: 12/08/2017. Board presentation: 12/08/2017

1. Prove that if $A$ and $B$ are finite sets, then $A\union B$ is a finite set.
2. 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.
3. Do Project 13.15, finding a formula for the bijection in the picture.
4. Prove Theorem 13.28.

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

1. 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.
2. Prove the converse of Prop. 11.2
3. Do Project 11.14
4. Prove that for all $x,y,z,w\in\R$ with $z,w\neq 0$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
5. 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 $\sqrt{2}$.
6. 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 $a_n$.