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- Previous Homework: 35 Hits
- h 330 - 02
**Previous****Homework**===== </WRAP> {{page>**people**:**fer**:**330ws**:**330ws**_**homework**_header&nofooter&noeditbtn}} ---- {{page>**people**:**fer**:**330ws**:defs&nofooter&noeditbtn}} **Problem Set 10** (complete) Due: 11/17/2017. Boa... is $m,n\in\N$ is removed. - Prove Prop. 2.38 ({{**people**:**fer**:**330ws**:appendix_ch2.pdf|appendix}}) - Prove Prop. 2.41.iii ({{**people**:**fer**:**330ws**:appendix_ch2.pdf|appendix}}) **Problem Set 2** (complete) Due: 09/08/2017. - Previous Homework: 26 Hits
- 330 - 03
**Homework**(Fall 2018)===== </WRAP> {{page>**people**:**fer**:**330ws**:**330ws**_**homework**_header&nofooter&noeditbtn}} ---- {{page>**people**:**fer**:**330ws**:defs&nofooter&noeditbtn}} **Problem Set 11** (complete) Due: 11/12/2018. Boa... presentation: 09/21/2018 - Prove Prop. 2.38 ({{**people**:**fer**:**330ws**:appendix_ch2.pdf|appendix}}) - Prove Prop. 2.41.iii ({{**people**:**fer**:**330ws**:appendix_ch2.pdf|appendix}}) **Problem Set 03** (complete) Due: 09/12/2018 - Homework: 25 Hits
- ~~META:title=
**Homework**~~ <WRAP centeralign> ===== Math 330 - 03**Homework**(Fall 2018)===== </WRAP> {{page>**people**:**fer**:**330ws**:**330ws**_**homework**_header&nofooter&noeditbtn}} ---- {{page>**people**:**fer**:**330ws**:defs&nofooter&noeditbtn}} **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 [[**people**:**fer**:**330ws**:fall2018:**previous**_**homework**|**Previous****Homework**]] [[**people**:**fer**:**330ws**:fall2018:home| Home]] - Homework: 24 Hits
- ~~META:title=
**Homework**~~ <WRAP centeralign> ===== Math 330 - 02**Homework**(Fall 2017)===== </WRAP> {{page>**people**:**fer**:**330ws**:**330ws**_**homework**_header&nofooter&noeditbtn}} ---- {{page>**people**:**fer**:**330ws**:defs&nofooter&noeditbtn}} **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 $a_n$. [[**people**:**fer**:**330ws**:fall2017:old_**homework**|**Previous****Homework**]] [[**people**:**fer**:**330ws**:fall2017:home| Home]]

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