people:fer:330ws:fall2018:previous_homework
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| + | ~~META:title=Previous 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 11** (complete) Due: 11/12/2018. Board presentation: 11/16/2018 | ||
| + | - Prove the following corollary to Prop. 10.4 \\ Corollary: $\glb(\R^+)=0$. | ||
| + | - Prove Prop. 10.7 | ||
| + | - Prove Prop. 10.10.iii | ||
| + | - Prove Prop. 10.13.ii | ||
| + | |||
| + | **Problem Set 10** (complete) Due: 11/05/2018. Board presentation: 11/14/2018 | ||
| + | - Let $f:A\to B$ and $g:B\to C$ be functions. | ||
| + | - If $g\circ f$ is injective, then $f$ is injective. | ||
| + | - If $g\circ f$ is surjective, then $g$ is surjective. | ||
| + | - Construct examples of functions $f:A\to B$ and $g:B\to C$ such that: | ||
| + | - $g\circ f$ is injective, but $g$ is not injective. | ||
| + | - $g\circ f$ is surjective, but $f$ is not surjective. | ||
| + | - Prove Prop. 9.15 (Hint: induction) | ||
| + | - Prove Prop. 9.18 | ||
| + | |||
| + | **Problem Set 09** (complete) Due: 10/29/2018. Board presentation: 11/05/2018 | ||
| + | -Prove Prop. 8.40.ii | ||
| + | -Prove Prop. 8.41 | ||
| + | -Prove Prop. 8.50 | ||
| + | -Give examples of subsets of $\R$ which are: | ||
| + | -bounded below and above, | ||
| + | -bounded below, but not bounded above, | ||
| + | -bounded above, but not bounded below, | ||
| + | -not bounded above or below. | ||
| + | |||
| + | **Problem Set 08** (complete) Due: 10/22/2018. Board presentation: 10/31/2018 | ||
| + | -Prove Prop. 6.16 | ||
| + | -Prove Prop. 6.17 | ||
| + | -Prove Prop. 6.25 (first part) | ||
| + | -Use Euclid's Lemma to prove the following corollary. Let $p$ be a prime, $k\in\N$, $m_1,m_2,\dots,m_k\in\N$. If $p|(m_1m_2\cdots m_k)$ then there is some $i$ with $1\leq i \leq k$ such that $p|m_i$. (Hint: Use induction on $k$). | ||
| + | |||
| + | **Problem Set 07** (complete) Due: 10/15/2018. Board presentation: 10/31/2018 | ||
| + | - Let $A$ be a set, and $\sim$ an equivalence relation on $A$. Let $A/\sim$ be the partition consisting of all equivalence classes of $\sim$. Let $\Theta_{(A/\sim)}$ be the equivalence relation induced by the partition $A/\sim$. Prove that $\Theta_{(A/\sim)}=\ \sim$. | ||
| + | - Do Project 6.8.iv. | ||
| + | |||
| + | **Problem Set 06** (complete) Due: 10/08/2018. Board presentation: 10/31/2018 | ||
| + | - Prove that set union is associative. | ||
| + | - Show, by counterexample, that set difference is not associative. | ||
| + | - Prove Prop. 5.20.ii | ||
| + | - Let $X$ and $Y$ be sets. Let $\power(X)$ denote the power set of $X$. Prove that: \[X\subseteq Y \iff \power(X)\subseteq\power(Y).\] | ||
| + | - (challenge) Prove that symmetric difference is associative. | ||
| + | |||
| + | **Problem Set 05** (complete) Due: 10/01/2018. Board presentation: 10/05/2018 | ||
| + | - Prove Prop. 4.6.iii | ||
| + | - Prove Prop. 4.11.ii | ||
| + | - Prove Prop. 4.15.i | ||
| + | - Prove Prop. 4.16.ii | ||
| + | |||
| + | **Problem Set 04** (complete) Due: 09/17/2018. Board 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. Board presentation: 09/17/2018 | ||
| + | - Prove that for all $k\in\N$, $k^2+k$ is divisible by 2. | ||
| + | - Prove Prop. 2.18.iii | ||
| + | - Prove Prop. 2.21. Hint: use proof by contradiction. | ||
| + | - Prove Prop. 2.23. Show, by counterexample, that the statement is not true if the hypothesis $m,n\in\N$ is removed. | ||
| + | - Fill-in the blank and prove that for all $k\geq\underline{\ \ }$, $k^2 < 2^k$. | ||
| + | |||
| + | **Problem Set 02** (complete) Due:09/05/2018. Board presentation: 09/10/2018 | ||
| + | - Prove Prop. 1.24 | ||
| + | - Prove Prop. 1.27.ii,iv | ||
| + | - Prove Prop. 2.7.i,ii | ||
| + | - Prove Prop. 2.12.iii | ||
| + | |||
| + | **Problem Set 01** (complete) Due: 08/27/2018. Board presentation: 08/31/2018 | ||
| + | - Prove Prop. 1.7 | ||
| + | - Prove Prop. 1.11.iv | ||
| + | - Prove Prop. 1.14 | ||
| + | - Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction).\\ Can you prove that 1 + 1 ≠ 0? | ||
| + | |||
| + | |||
| + | [[people:fer:330ws:fall2018:home| Home]] | ||