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## Math 330 - 02 Previous Homework

• LaTeX-ed solutions are encouraged and appreciated.
• If you use LaTeX, hand-in a printed version of your homework.
• You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange of any written material.
• Writing of homework problems should be done on an individual basis.
• References to results from the textbook and/or class notes should be included.
• The following lists should be considered partial and tentative lists until the word complete appears next to it.
• Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.


Problem Set 10 (complete) Due: 11/17/2017. Board Presentation: 11/17/2017

1. Prove Prop. 10.23.v
2. Prove The. 10.26
3. 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 $L$.
4. 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

1. Prove Prop. 10.10.iii
2. Prove Prop. 10.17
3. Prove Prop. 10.23.iii

Problem Set 8 (complete) Due: 11/03/2017. Board presentation: 11/03/2017

1. Prove Prop. 8.50
2. Prove that function composition is associative, when defined.
3. 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.
4. Construct an example of a function with several right inverses.
5. Prove Prop. 9.15 (Hint: induction on $k$)
6. Prove Prop. 9.18

Problem Set 7 (complete) Due: 10/27/2017. Board presentation: 10/27/2017

1. 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 $k$)
2. Prove Prop. 8.6
3. Prove Prop. 8.40.ii
4. Prove Prop. 8.41

Problem Set 6 (complete) Due: 10/13/2017. Board presentation: 10/20/2017

1. Let $f_n$ be the $n$-th Fibonacci number. Prove by induction on $n$ that $\sum_{j=1}^n f_{2j} = f_{2n+1}-1$
2. Find and write down all the partitions on a 4-element set $A=\{a,b,c,d\}$. How many equivalence relations are there on $A$?
3. Prove Prop. 6.15
4. Prove Prop. 6.16

Problem Set 5 (complete) Due: 10/06/2017. Board presentation: 10/18/2017

1. 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.
2. 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.
3. Prove the following identities for the Fibonacci numbers $f_{2n+1}=f_n^2+f_{n+1}^2;\quad \\ f_{2n}=f_{n+1}^2-f_{n-1}^2 = f_n(f_{n+1}+f_{n-1})$
4. 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

1. Prove Prop. 4.6.iii
2. Prove Prop. 4.11.ii
3. Do project 4.12
4. Prove Prop. 4.16.ii

Problem Set 3 (complete) Due: 09/15/2017. Board presentation: 09/20/2017

1. Prove Prop. 2.21 (Hint: proof by contradiction)
2. Prove Prop. 2.23. Show, by counterexample, that the statement is not true when the hypothesis $m,n\in\N$ is removed.
3. Prove Prop. 2.38 (appendix)
4. Prove Prop. 2.41.iii (appendix)

Problem Set 2 (complete) Due: 09/08/2017. Board presentation: 09/15/2017

1. Prove Prop. 1.25
2. Prove Prop. 1.27.iv
3. Prove Prop. 2.7
4. Prove transitivity of $"\leq"$.

Problem Set 1 (complete) Due: 09/01/2017. Board Presentation: 09/08/2017

1. Prove Prop. 1.7
2. Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction). Can you prove that 1 + 1 ≠ 0?
3. Prove Prop. 1.11.iv
4. Prove Prop. 1.14