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===== Math 330 - 03 Homework (Fall 2018)=====
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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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