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===== Math 330 - 02 Previous Homework =====
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**Problem Set 10** (complete) Due: 11/17/2017. Board Presentation: 11/17/2017
  - Prove Prop. 10.23.v
  - Prove The. 10.26
  - Let $(a_n)$ be a sequence. Consider the sequence of even-indexed terms, $(a_{2n})$, and the sequence of odd-indexed terms, $(a_{2n+1})$.  Prove that if both $(a_{2n})$ and $(a_{2n+1})$ converge to $L$, then $(a_n)$ converges to $L$. 
  - Let $q_n=\displaystyle\frac{f_n}{f_{n+1}}$, where $f_n$ is the $n$-th Fibonacci number.  Show that the sequence $(q_n)$ converges.  What value does it converge to?

**Problem Set 9** (complete) Due: 11/10/2017. Board presentation: 11/10/2017
  - Prove Prop. 10.10.iii
  - Prove Prop. 10.17
  - Prove Prop. 10.23.iii

**Problem Set 8** (complete) Due: 11/03/2017. Board presentation: 11/03/2017
  - Prove Prop. 8.50
  - Prove that function composition is associative, when defined.
  - Let $A,B,C$ be sets and $f:A\to B$ and $g:B\to C$ functions. Prove that if $g\circ f$ is surjective then $g$ is surjective. Give an example when $g\circ f$ is surjective, but $f$ is not.
  - Construct an example of a function with several right inverses. 
  - Prove Prop. 9.15 (Hint: induction on $k$)
  - Prove Prop. 9.18

**Problem Set 7** (complete) Due: 10/27/2017. Board presentation: 10/27/2017
  - Prove the corollary to Prop. 6.25: Let $a,b\in\Z$, $n\in\N$ and $k\geq 0$.  If $a \equiv b \pmod{n}$ then $a^k \equiv b^k \pmod{n}$. (Hint: induction on $k$)
  - Prove Prop. 8.6
  - Prove Prop. 8.40.ii
  - Prove Prop. 8.41

**Problem Set 6** (complete) Due: 10/13/2017. Board presentation: 10/20/2017
  - Let $f_n$ be the $n$-th Fibonacci number. Prove by induction on $n$ that \[ \sum_{j=1}^n f_{2j} = f_{2n+1}-1 \]
  - Find and write down all the partitions on a 4-element set $A=\{a,b,c,d\}$. How many equivalence relations are there on $A$?
  - Prove Prop. 6.15
  - Prove Prop. 6.16

**Problem Set 5** (complete) Due: 10/06/2017. Board presentation: 10/18/2017
  - Let $n\in\N$.  Prove that if $n$ is divisible by 3, then $f_n$ is even. Is the converse true? If so, prove it; if not, give a counterexample.
  - Let $n\in\N$.  Prove that if $n$ is divisible by 5, then $f_n$ is divisible by 5. Is the converse true? If so, prove it; if not, give a counterexample.
  - Prove the following identities for the Fibonacci numbers \[ f_{2n+1}=f_n^2+f_{n+1}^2;\quad \\ f_{2n}=f_{n+1}^2-f_{n-1}^2 = f_n(f_{n+1}+f_{n-1}) \]
  - Prove the associativity of the set union and set intersection operations.  Give a counterexample to show that set difference is not associative. 

**Problem Set 4** (complete) Due: 09/29/2017.  Board presentation: 10/06/2017
  - Prove Prop. 4.6.iii
  - Prove Prop. 4.11.ii
  - Do project 4.12
  - Prove Prop. 4.16.ii


**Problem Set 3** (complete) Due: 09/15/2017.  Board presentation: 09/20/2017
  - Prove Prop. 2.21 (Hint: proof by contradiction)
  - Prove Prop. 2.23. Show, by counterexample, that the statement is not true when the hypothesis $m,n\in\N$ 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.  Board presentation: 09/15/2017
  - Prove Prop. 1.25
  - Prove Prop. 1.27.iv
  - Prove Prop. 2.7
  - Prove transitivity of $"\leq"$.

**Problem Set 1** (complete) Due: 09/01/2017.  Board Presentation: 09/08/2017
  - Prove Prop. 1.7
  - Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction). Can you prove that 1 + 1 ≠ 0?
  - Prove Prop. 1.11.iv
  - Prove Prop. 1.14 



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