people:fer:330ws:spring2022:homework

Math 330 - 01 Homework (Spring 2022)

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 13 (complete) Due: 05/09/2022

Let $f:A\to B$ and $g:C\to D$ be functions. Define $f\times g:A\times C \to B\times D$ by $(f\times g)(a,c)=(f(a),g(c))$ .
Prove that if $f$ and $g$ are surjective, then so is $f\times g$ .
Prove that the function $\ f:\Z \to \N$ given by $f(m) = \cases {2m &if $m>0,$ \cr -2m+1 &if $m\leq 0,$ \cr}$ is bijective.
Prove that if $A$ and $B$ are finite sets, then so is $A\union B$ . Morevoer, if $A$ and $B$ are disjoint, then $|A\union B|=|A|+|B|$ .
Prove Theorem 13.28. Hint: consider the function $\tan(x)$ from calculus.

Problem Set 12 (complete) Due: 05/02/2022. Board presentation: 05/06/2022

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 $\sqrt{2}$ .
Prove Prop. 11.21.iii

Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022

Prove part (iv) of lemma stated in class:
for $x\in\R$ and $r\in\R^+$ ,
(iv) $|x| \leq r$ iff $x \leq r$ and $-x \leq r$ .
(Hint: use part (iii) of the same lemma.
Prove Prop. 10.10.iii (Hint: use 10.8.iv)
Prove Prop. 10.13.ii
Prove Prop. 10.17 (Hint: use induction)

Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022

Let $f:A\to B$ and $g:B\to C$ be functions.
1. Prove Prop. 9.7.ii
2. Prove that if $g\circ f$ is surjective, then $g$ is surjective.
Prove Prop. 9.10.ii
Prove Prop. 9.15 (Hint: induction)
Prove Prop. 9.18

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