Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 1

Warm-up-exercises

Exercise

Let ${}A,B$ and ${}C$ denote sets. Prove the following identities.

1. ${}A\cup \emptyset =A\,,$

2. ${}A\cap \emptyset =\emptyset \,,$

3. ${}A\cap B=B\cap A\,,$

4. ${}A\cup B=B\cup A\,,$

5. ${}A\cap (B\cap C)=(A\cap B)\cap C\,,$

6. ${}A\cup (B\cup C)=(A\cup B)\cup C\,,$

7. ${}A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\,,$

8. ${}A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\,,$

9. ${}A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)\,.$

Exercise

Prove the following (set-theoretical versions of) syllogisms of Aristotle. Let ${}A,B,C$ denote sets.

1. Modus Barbara: ${}B\subseteq A$ and ${}C\subseteq B$ imply ${}C\subseteq A$.
2. Modus Celarent: ${}B\cap A=\emptyset$ and ${}C\subseteq B$ imply ${}C\cap A=\emptyset$.
3. Modus Darii: ${}B\subseteq A$ and ${}C\cap B\neq \emptyset$ imply ${}C\cap A\neq \emptyset$.
4. Modus Ferio: ${}B\cap A=\emptyset$ and ${}C\cap B\neq \emptyset$ imply ${}C\not \subseteq A$.
5. Modus Baroco: ${}B\subseteq A$ and ${}B\not \subseteq C$ imply ${}A\not \subseteq C$.

Exercise

Prove the following formulas by induction.

1. ${}\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\,,$

2. ${}\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}\,.$

3. ${}\sum _{i=1}^{n}i^{3}={\left({\frac {n(n+1)}{2}}\right)}^{2}\,.$

Exercise

Show that (with ${}n=3$ being the only exception) the relation

${}2^{n}\geq n^{2}\,$

holds.

Exercise *

Show, by induction, that for every ${}n\in \mathbb {N}$, the number

$6^{n+2}+7^{2n+1}$

is a multiple of ${}43$.

Exercise

Prove, by induction, that the following inequality holds

${}1\cdot 2^{2}\cdot 3^{3}\cdots n^{n}\leq n^{\frac {n(n+1)}{2}}\,.$

Exercise *

Prove, by induction, that the formula

${}\sum _{k=1}^{n}(-1)^{k-1}k^{2}=(-1)^{n+1}{\frac {n(n+1)}{2}}\,$

holds for all ${}n\in \mathbb {N} _{+}$.

Exercise

The cities ${}S_{1},\ldots ,S_{n}$ are connected by roads, and there is exactly one road between each couple of cities. Due to construction works, at the moment all roads are drivable only in one direction. Show that nevertheless, there exists one city from which you can reach all the others.

Hand-in-exercises

Exercise (4 marks)

Let ${}A$ and ${}B$ be sets. Show that the following facts are equivalent.

1. ${}A\subseteq B$,
2. ${}A\cap B=A$
3. ${}A\cup B=B$,
4. ${}A\setminus B=\emptyset$,
5. There exists a set ${}C$ such that ${}B=A\cup C$,
6. There exists a set ${}D$ such that ${}A=B\cap D$.

Exercise * (3 marks)

Prove, by induction, that the sum of consecutive odd numbers (starting from ${}1$) is always a square number.

Exercise (3 marks)

Fix ${}m\in \mathbb {N}$. Show, by induction, that the following identity holds.

${}(2m+1)\prod _{i=1}^{m}(2i-1)^{2}=\prod _{k=1}^{m}(4k^{2}-1)\,.$

Exercise (4 marks)

An ${}n$-chocolate is a rectangular grid, which is divided by ${}a-1$ longitudinal grooves and by ${}b-1$ transverse grooves into ${}n=a\cdot b$ smaller bite-sized rectangles. A dividing step of a chocolate is the complete severing of a chocolate, along a longitudinal or a transverse groove. A complete breakdown of a chocolate is a consequence of division steps (each one applied to a previously obtained intermediate chocolate), whose final product consists of all the small bite-sized pieces, more handy to be eaten. Show, by induction, that each breakdown of an ${}n$-chocolate consists of exactly ${}n-1$ division steps.