~~META:title=Previous Homework~~

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===== Math 330 - 03 Homework (Fall 2018)=====
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**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?


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