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{\bf \Large Math 112 Homework 1}\medskip

Due Thursday February 7, 2019, {\bf on Canvas}.
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You are encouraged to discuss the homework problems with 
other students. However, what you hand in should reflect your 
own understanding of the material. You are NOT allowed to copy solutions 
from other students or other sources. {\it No late homeworks will be 
accepted.}

Please pay attention to the clarity and precision of your answers. Your
solutions to the problems should always consist of carefully written 
mathematical arguments. 

Be especially careful about making the structure of your argument
understandable: use sentences (``Assume that ...'', ``We want to prove that ...'',
``Therefore, ...'') to clarify the status of the various mathematical
assertions in your answer.
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{\bf Material covered:} Rudin pages 1-17, 24-30.
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{\bf 1.} (3 points) Rudin Chapter 1 Problem 1.

{\bf 2.} (5 points) Rudin Chapter 1 Problem 5.

{\bf 3.} (5 points) Rudin Chapter 1 Problem 9.
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{\bf 4.} (5 points)

(a) Let $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}: a,b\in\mathbb{Q}\}$. Prove that
$\mathbb{Q}(\sqrt{2})$ is a subfield of $\mathbb{R}$.

(Hint: observe that, since $\mathbb{Q}(\sqrt{2})$ is contained in 
$\mathbb{R}$, it is enough to check that the operations $x+y$, $-x$, $xy$, 
and $1/x$ are well-defined in $\mathbb{Q}(\sqrt{2})$ -- explain why).

(b) Prove that $\sqrt{3}$ does not belong to $\mathbb{Q}(\sqrt{2})$.

(Hint: compute the square of $a+b\sqrt{2}$ and argue by contradiction).
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{\bf 5.} (3 points)

Assume that $n\ge 2$ is an integer and $z_1,z_2,\dots,z_n$ are complex
numbers such that $z_1+z_2+\dots+z_n=0$ and $|z_1|=|z_2|=\dots=|z_n|=1$.
Prove that $$|1+z_1|^2+|1+z_2|^2+\dots+|1+z_n|^2=2n.$$

{\bf 6.} (4 points) Rudin Chapter 1 Problem 18.
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{\bf 7.} (5 points)
Let $\mathbb{Z}$ be the set of all integers, and $M=\{S\subset \mathbb{Z},\ 
S$ is finite$\}$ the set of all finite subsets of $\mathbb{Z}$. Show that
$M$ is countable.

(Hint: show that $M$ can be described as the union of countably many 
finite or countable sets).
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{\bf 8.} How long did this assignment take you?  How hard was it?
What resources did you use, and how much help did you need?
Did you have any prior experience with abstract math and writing 
mathematical arguments?

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