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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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===== Math 330 - 01 Homework (Spring 2022)=====
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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.
- Prove Prop. 9.7.ii
- 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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===== Math 330 - 02 Homework (Fall 2017)=====
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**Problem Set 12** (complete) Due: 12/08/2017. Board presentation: 12/08/2017
- Prove that if $A$ and $B$ are finite sets, then $A\union B$ is a finite set.
- Prove the following corollary to Proposition 13.6.
- If $f:A\to B$ is injective and $B$ is finite, then $A$ is finite.
- If $g:A\to B$ is surjective and $A$ is finite, then $B$ is finite.
- Do Project 13.15, finding a formula for the bijection in the picture.
- Prove Theorem 13.28.
**Problem Set 11** (complete) Due: 12/01/2017. Board Presentation: 12/01/2017
- 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.
- Prove the converse of Prop. 11.2
- Do Project 11.14
- Prove that for all $x,y,z,w\in\R$ with $z,w\neq 0$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
- 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 $a_n$.
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