# CMSC56 2nd Long Exam
Deadline: December 6
Write your answers in a blue book and submit it to my table

Write your solutions clearly. I'll be more inclined to give effort points to solutions I understand.

## I. Proofs

1. Prove that if $A \subseteq B$ then $\overline{A} \cup B=U$ where $U$ is the universal set and $A$ and $B$ are arbitrary sets.

2. Prove that the closed form sum of perfect cubes is true for all possible natural numbers, $n$. Use proof by induction.
$$
\sum_{i=0}^{n}{i^2}=\frac{n(n+1)(2n+1)}{6}
$$

3. Prove that the closed form sum of perfect cubes is true for all possible natural numbers, $n$. Use proof by induction.

$$
\sum_{i=0}^{n}{i^3}=\frac{n^2(n+1)^2}{4}
$$

4. Prove that the the following statement is true for all natural numbers $n$. Use proof by Induction.

$$
11^n-6 \text{ is divisible by 5}
$$

5. Prove that the difference of a rational number minus an irrational number is always an irrational number. Use proof by contradiction.

## II. Sets

#### A. Describe the following sets using set builder notation

1. The set of all sets containing the element 1
2. The set of all perfect squares divisible by 3
3. The set of all integers with less than three digits when written in base-10
4. The set of all natural numbers not divisible by 11
5. The set of all natural numbers with even square roots

#### B. Shade the regions of a venn diagram of the corresponding sets:

1. $(A - B) \cap C$
2. $(B \cup C)\cap(A-B)$
3. $A \cup (B \cap C)$

## III. Functions

##### A. Indicate whether these functions are surjective, injective, neither, or both (bijective). If the function is not bijective, change either the domain and range to make it bijective

1. $f:R \to R$ where $f(x)=\frac{1}{x}$
2. $g:R \to R$ where $g(x)=x^3$
3. $h: R \to R$ where $h(x)=\lceil x\rceil-x$
4. $i:R \to R$ where $i(x)=\log x$
5. $j:R\to R$ where $j(x)=\sin x$

##### B. Write the inverses of all of the the bijections you wrote from each item in test A. (since each of those numbers are either bijective or changed to become bijective, all 5 items will have an inverse)

##### C. Given $k(x)=x^3$, $l(x)=-\frac{4}{x}$, and $m(x)=5-2x$, solve for the following functions

1. $kl-m$
2. $k+m-l^{-1}$
3. $m^{-1}k^{-1}$

## IV. Sequences and summations

#### A. Write down the first 10 numbers of these sequences

1. $\{a_n\}$ where $a_n=4+5n$
2. $\{b_n\}$ where $b_n=3(-2)^n$
3. $\{c_n\}$ where $c_0=4$ and $c_n=c_{n-1}-4n$

4. $\{d_n\}$ where $d_0=5$ and $d_n = 2d_{n-1} + 6$
5. $\{e_n\}$ where $e_0=2$ and $e_n=5e_{n-1}+1$

#### B. Solve for the closed for formula of the following sequences from part A

1. $d_n$
2. $e_n$

#### C. Solve for following summations based on the sequences from part A

1. $\sum_{i=0}^{100}{a_i}$
2. $\sum_{i=0}^{10}{b_i}$

D. Solve for the following nested sums

1. $\sum_{i=0}^{100}({\sum_{j=0}^{100}{ij}})$
2. $\sum_{i=0}^{100}({\sum_{j=i}^{100}{j}})$