~~META:title=Previous Homework~~

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===== Math 330 - 01 Homework (Spring 2022)=====
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=== Previous Homework ===
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**Problem Set 09** (complete) Due: 04/04/2022. Board presentation: 04/08/2022
  -Prove Prop. 8.40.ii
  -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. 
  -Project 9.3

**Problem Set 08** (complete) Due: 03/28/2022. Board presentation:  04/01/2022
  -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 $k$).
  -Prove Prop. 6.28

**Problem Set 07** (complete)  Due: 03/21/2022.  Board presentation: 03/25-28/2022
  - Prove that set union is associative.
  - Prove Prop. 5.20.ii
  - Let $X$ and $Y$ be sets. Let $\power(X)$ denote the power set of $X$. Prove that: \[X\subseteq Y \iff \power(X)\subseteq\power(Y).\]
  - 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 the two equivalence relations $\Theta_{(A/{\sim})}$ and $\sim$ are equal. 

**Problem Set 06** (complete)  Due: 03/07/2022.  Board presentation: 03/11/2022
  - Prove that $$\sum_{k=2}^n \binom{k}{2} = \binom{n+1}{3}$$ Hint: use induction on $n$.
  - Prove that for $k\geq 1$, $$\sum_{m=0}^k (-1)^m \binom{k}{m} = 0$$
  - Determine the base case, and prove by induction and using the recursive definition of Fibonacci numbers that $$f_{2k}=f_{k+1}^2-f_{k-1}^2$$ 
  - Prove Prop. 4.31 without using Prop. 4.29

**Problem Set 05** (complete) Due: 02/28/2022.  Board presentation: 03/04/2022
  - Prove Prop. 4.6.iii
  - Prove Prop. 4.11.ii
  - Prove Prop. 4.15.i
  - Prove Prop. 4.18

**Problem Set 04** (complete) Due: 02/21/2022.  Board presentation: 02/28/2022
  - Prove Prop. 2.38  ({{people:fer:330ws:appendix_ch2.pdf|appendix}})
  - Prove Prop. 2.41.iii ({{people:fer:330ws:appendix_ch2.pdf|appendix}})
  - Project 3.1

**Problem Set 03** (complete) Due: 02/14/2022.  Board presentation: 02/18/2022
  - Prove that for all $k\in\N$, $k^2+k$ is divisible by 2. Is this true for all $k\in\Z$?
  - 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:02/07/2022.  Board presentation: 02/11/2022
  - 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: 01/31/2022. Board presentation: 02/04/2022 (rescheduled for 02/07/2022)
  - 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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