Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 4

Exercises

Exercise

Show that the operation on a line that assigns to two points their midpoint, is commutative, but not associative. Does there exist a neutral element?

Exercise

Let ${}M$ be the set of all female compound first names (names of the form ${}A-B$, where ${}A,B$ are simple first names and every combination is allowed). We consider the composition

$M\times M\longrightarrow M$

which sends a pair of double first names ${}(A-B,C-D)$ to the double name ${}A-D$.

1. What is the value of ${}({\text{Lea-Marie}},{\text{Klara-Sophie}})$ under this composition?
2. Is this composition commutative?
3. Is this composition associative?
4. Does this composition have a neutral element?
5. Is this composition surjective?
6. Is this composition injective?

Exercise

Let ${}x\neq 0$ be an element in a field ${}K$. Show that the element ${}z$ fulfilling ${}xz=1$ is uniquely determined.

Exercise

Let ${}K$ be a field and let ${}a,b,c$ be elements from ${}K$. Show that the following rules for sign hold.

1. ${}(-a)b=-ab=a(-b)\,.$

2. ${}(-a)(-b)=ab\,.$

3. ${}a(b-c)=ab-ac\,.$

Exercise

Describe and prove rules for the addition and the multiplication of even and odd integer numbers. Define on the set with two elements

$\{E,O\}$

an "addition“ and a "multiplication“ that "represent“ these rules.

Exercise

Let ${}K$ be a field. Show that, for every natural number ${}n\in \mathbb {N}$, there exists a field element ${}n_{K}$ such that ${}0_{K}$ is the null element in ${}K$, and ${}1_{K}$ is the unit element in ${}K$, and such that

${}(n+1)_{K}=n_{K}+1_{K}\,$

holds. Show that this assignment has the properties

$(n+m)_{K}=n_{K}+m_{K}{\text{ and }}(nm)_{K}=n_{K}\cdot m_{K}.$

Extend this assignment to all integer numbers ${}\mathbb {Z}$, and show that the stated structural properties hold again.

Exercise

Let ${}K$ be a field with ${}2\neq 0$. Show that for ${}f,g\in K$, the relation

${}fg={\frac {1}{4}}{\left({\left(f+g\right)}^{2}-{\left(f-g\right)}^{2}\right)}\,$

holds.

Exercise

Let ${}x,y,z,w$ be elements in a field, and suppose that ${}z$ and ${}w$ are not zero. Prove the following fraction rules.

1. ${}{\frac {x}{1}}=x\,,$

2. ${}{\frac {1}{z}}=z^{-1}\,,$

3. ${}{\frac {1}{-1}}=-1\,,$

4. ${}{\frac {0}{z}}=0\,,$

5. ${}{\frac {z}{z}}=1\,,$

6. ${}{\frac {x}{z}}={\frac {xw}{zw}}\,$

7. ${}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$

8. ${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$

Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is

${}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?$

Show that the popular formula

${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,$

does not hold.

Exercise

Let ${}K$ be a field and let ${}a,b\neq 0$ be elements in ${}K$. Prove the following laws for powers for natural exponents ${}m,n\in \mathbb {N}$.

1. ${}a^{m+n}=a^{m}\cdot a^{n}\,.$

2. ${}{\left(a^{m}\right)}^{n}=a^{mn}\,.$

3. ${}(a\cdot b)^{n}=a^{n}\cdot b^{n}\,.$

Exercise

Let ${}K$ be a field and let ${}a,b\neq 0$ be elements in ${}K$. Prove the following laws for powers for integer exponents ${}m,n\in \mathbb {Z}$. The corresponding laws for exponents from ${}\mathbb {N}$ can be used and also the facts that the inverse of the inverse is the element itself and that the inverse of ${}u^{k}$ equals ${}{\left(u^{-1}\right)}^{k}$.

1. ${}a^{m+n}=a^{m}\cdot a^{n}\,.$

2. ${}{\left(a^{m}\right)}^{n}=a^{mn}\,.$

3. ${}(a\cdot b)^{n}=a^{n}\cdot b^{n}\,.$

Exercise

Compute the matrix product

${\begin{pmatrix}0&1\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}1&0\\0&0\end{pmatrix}}.$

Exercise *

a) Give an example of rational numbers ${}a,b,c\in {]0,1[}$ such that

${}a^{2}+b^{2}=c^{2}\,.$

b) Give an example of rational numbers ${}a,b,c\in {]0,1[}$ such that

${}a^{2}+b^{2}\neq c^{2}\,.$

c) Give an example of irrational numbers ${}a,b\in {]0,1[}$ and a rational number ${}c\in {]0,1[}$ such that

${}a^{2}+b^{2}=c^{2}\,.$

Exercise

Show that the binomial coefficients satisfy the following recursive relation

${}{\binom {n+1}{k}}={\binom {n}{k}}+{\binom {n}{k-1}}\,.$

Exercise

Show that the binomial coefficients are natural numbers.

Exercise

Prove the formula

${}2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}\,.$

Exercise *

Show by induction that for ${}n\geq 10$ the estimate

${}3^{n}\geq n^{4}\,$

holds.

Exercise

Franziska wants to break up with her boyfriend Heinz. She thinks about the following explanations.

1. "You behaved already on the very first day completely weird. Since then, from each day to the next day, you got worse. Hence, you will continue to behave completely weird“.
2. "If I would stay together with you, I will end up as a sad bored person disappointed with life, and this I definitely do not want“.
3. "Well, if you do not love me, then I do not want you anyway. But if you love me, then I come to the conclusion that you are not able to behave in a way which fits your feelings. But then you are not mature, and then I do not want you neither“.

Which mathematical proof principles can be seen in these explanations?

Exercise

1. Solve the following mini-sudoku

   ${\begin{pmatrix}-&-&2&-\\3&-&-&4\\-&-&-&-\\-&4&-&1\end{pmatrix}}.$

2. Show that the mini-sudoku in (1) has only one solution.
3. Which mathematical proof principles can be found as typical argumentation patterns when solving a sudoku?

Exercise

Look at the following statement: "The principle "Proof by contradiction“ is obviously nonsense. If we can assume anything, then we can always arrive at a contradiction and then we can prove anything“. Discuss this statement.

Hand-in-exercises

Exercise (2 marks)

Show that taking powers of positive natural numbers, i.e., the assignment

$\mathbb {N} \times \mathbb {N} \longrightarrow \mathbb {N} ,(a,b)\longmapsto a^{b},$

is neither commutative nor associative. Does this operation have a neutral element?

Exercise (2 marks)

Let ${}a$ be an element in a field different from ${}0$. Show how to compute ${}a^{10}$ with four multiplications.

Exercise (5 marks)

Prove the general distributive law for a field.

Exercise (3 marks)

Show that the "rule“

${}{\frac {a}{b}}+{\frac {c}{d}}={\frac {a+c}{b+d}}\,$

is for ${}a,c\in \mathbb {N} _{+}$ (and ${}b,d,b+d\in \mathbb {Z} \setminus \{0\}$) never true. Give an example with ${}a,b,c,d,b+d\neq 0$ where this rule holds.

Exercise (2 (1+1) marks)

Let ${}u\in \mathbb {R}$ and ${}v\in \mathbb {Q}$.

1. Show that ${}u$ is an irrational number if and only if ${}u+v$ is irrational.
2. Suppose now also that ${}v\neq 0$. Show that ${}u$ is irrational if and only if ${}u\cdot v$ is irrational.

Exercise (3 marks)

Prove the following formula

${}n2^{n-1}=\sum _{k=0}^{n}k{\binom {n}{k}}\,.$

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