Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 3/refcontrol

Exercises

Exercise Create referencenumber

Determine, for the sets

$M=\{a,b,c,d,e\},\,N=\{a,c,e\},\,P=\{b\},\,R=\{b,d,e,f\},$

the following sets.

1. ${}M\cap N$,
2. ${}M\cap N\cap P\cap R$,
3. ${}M\cup R$,
4. ${}{\left(N\cup P\right)}\cap R$,
5. ${}N\setminus R$,
6. ${}{\left(M\cup P\right)}\setminus {\left(R\setminus N\right)}$,
7. ${}{\left({\left(P\cup R\right)}\cap N\right)}\cap R$,
8. ${}{\left(R\setminus P\right)}\cap {\left(M\setminus N\right)}$.

Exercise Create referencenumber

Let ${}LA$ denote the set of capital letters in the Latin alphabet, ${}GA$ the set of capital letters in the Greek alphabet, and ${}RA$ the set of capital letters in the Russian alphabet. Determine the following sets.

1. ${}GA\setminus RA$.
2. ${}{\left(LA\cap GA\right)}\cup {\left(LA\cap RA\right)}$.
3. ${}RA\setminus {\left(GA\cup RA\right)}$.
4. ${}RA\setminus {\left(GA\cup LA\right)}$.
5. ${}{\left(RA\setminus GA\right)}\cap {\left({\left(LA\cup GA\right)}\setminus {\left(GA\cap RA\right)}\right)}$.

Exercise Create referencenumber

Let ${}A,\,B$ and ${}C$ denote sets. Prove the identity

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

Exercise Create referencenumber

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 Create referencenumber

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 Create referencenumber

Does the "subtraction rule“ hold for the union of sets, i.e., can we infer from ${}A\cup C=B\cup C$ that ${}A=B$ holds?

Exercise Create referencenumber

Let ${}A,B,C$ denote sets. Show that the following statements are equivalent.

1. ${}A\subseteq B\cup C$.
2. ${}A\setminus B\subseteq C$
3. ${}A\setminus C\subseteq B$.

Exercise Create referencenumber

Sketch the product setMDLD/product set ${}\mathbb {N} \times \mathbb {N}$ as a subset of ${}\mathbb {R} \times \mathbb {R}$.

Exercise Create referencenumber

Describe for any choice of two of the following geometric sets its product set (including the case where a set is taken twice).

1. A line segment ${}I$.
2. A circle (circumference) ${}K$.
3. A disk ${}D$.
4. A parabola ${}P$.

Which of these product sets can be realized within space, which can't?

Exercise Create referencenumber

Sketch the following subsets in ${}\mathbb {R} ^{2}$.

1. ${}{\left\{(x,y)\mid x=7{\text{ or }}y=3\right\}}$,
2. ${}{\left\{(x,y)\mid 7x\geq 3y{\text{ and }}4x\leq y\right\}}$,
3. ${}{\left\{(x,y)\mid x^{2}+y^{2}=0\right\}}$,
4. ${}{\left\{(x,y)\mid x^{2}+y^{2}=1\right\}}$.

Exercise Create referencenumber

1. Sketch the set ${}M={\left\{(x,y)\in \mathbb {R} ^{2}\mid 4x-7y=3\right\}}$ and the set ${}N={\left\{(x,y)\in \mathbb {R} ^{2}\mid 3x+2y=5\right\}}$.
2. Determine the intersection ${}M\cap N$ geometrically and computationally.

We recommend illustrating the formulated set identities of the following exercises.

Exercise Create referencenumber

Let ${}M$ and ${}N$ denote sets, and let ${}A\subseteq M$ and ${}B\subseteq N$ be subsets. Show the identity

${}{\left(A\times N\right)}\cap {\left(M\times B\right)}=A\times B\,.$

Exercise Create referencenumber

Let ${}M$ and ${}N$ denote sets, and let ${}A_{1},A_{2}\subseteq M$ and ${}B_{1},B_{2}\subseteq N$ be subsets. Show the identity

${}{\left(A_{1}\times B_{1}\right)}\cap {\left(A_{2}\times B_{2}\right)}={\left(A_{1}\cap A_{2}\right)}\times {\left(B_{1}\cap B_{2}\right)}\,.$

Exercise Create referencenumber

Let ${}P$ be a set of people and ${}V$ the set of the first names and ${}N$ the set of the surnames of these people. Define natural mappings from ${}P$ to ${}V$, to ${}N$ and to ${}V\times N$ and discuss them using the relevant notions for mappings.

Exercise Create referencenumber

Dr. Peter Klamser

Determine for the following diagrams which empirical mappings they describe. What is in each case the source, the target, which units are used? Does each image represent one or more mappings? Do they really represent mappings? What information is conveyed beyond the mapping? Is there some kind of mathematical modelling for these empirical mappings?

Exercise Create referencenumber

Give examples of mappingsMDLD/mappings

$\varphi ,\psi \colon \mathbb {N} \longrightarrow \mathbb {N}$

such that ${}\varphi$ is injectiveMDLD/injective but not surjective,MDLD/surjective and ${}\psi$ is surjective but not injective.

Exercise Create referencenumber

Show that there exists a bijectionMDLD/bijection between ${}\mathbb {N}$ and ${}\mathbb {Z}$.

Exercise Create referencenumber

Establish, for each ${}n\in \mathbb {N}$, whether the function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

is injectiveMDLD/injective and/or surjective.MDLD/surjective

Exercise Create referencenumber

Which graphsMDLD/graphs (map) of a function ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ do you know from school?

Exercise Create referencenumber

How can we recognize by looking at the graphMDLD/graph (map) of a mapping

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

whether ${}f$ is injectiveMDLD/injective or surjective?MDLD/surjective

Exercise Create referencenumber

Which bijectiveMDLD/bijective functions ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ (or between subsets of ${}\mathbb {R}$) do you know from school? What is the name of the inverse function?MDLD/inverse function

Exercise Create referencenumber

Let ${}L$ and ${}M$ denote sets. Show that the mapping

$\tau \colon L\times M\longrightarrow M\times L,(x,y)\longmapsto (y,x),$

is a bijective mappingMDLD/bijective mapping between the product sets ${}L\times M$ and ${}M\times L$.

Exercise Create referencenumber

Let ${}L$ and ${}M$ be sets and let

$F\colon L\longrightarrow M$

be a function. Let

$G\colon M\longrightarrow L$

be another function such that ${}F\circ G=\operatorname {Id} _{M}\,$ and ${}G\circ F=\operatorname {Id} _{L}\,$. Show that ${}G$ is the inverseMDLD/inverse (map) of ${}F$.

Exercise Create referencenumber

We consider the sets

$L=\{1,2,3,4,5,6,7,8\},\,M=\{a,b,c,d,e,f,g,h,i\}{\text{ and }}N=\{R,S,T,U,V,W,X,Y,Z\}$

and the mappings ${}\varphi \colon L\rightarrow M$ and ${}\psi \colon M\rightarrow N$ which are given by the value tables

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$	${}8$
${}\varphi (x)$	${}c$	${}i$	${}a$	${}g$	${}d$	${}e$	${}h$	${}b$

and

${}y$	${}a$	${}b$	${}c$	${}d$	${}e$	${}f$	${}g$	${}h$	${}i$
${}\psi (y)$	${}X$	${}Z$	${}Y$	${}S$	${}Z$	${}S$	${}T$	${}W$	${}U$

respectively.

1. Determine a value table for ${}\psi \circ \varphi$.
2. Are the mappings ${}\varphi$, ${}\psi$, ${}\psi \circ \varphi$ injective?
3. Are the mappings ${}\varphi$, ${}\psi$, ${}\psi \circ \varphi$ surjective?

Exercise Create referencenumber

Determine the composite functionsMDLD/composite functions ${}\varphi \circ \psi$ and ${}\psi \circ \varphi$ for the functionsMDLD/functions ${}\varphi ,\psi \colon \mathbb {R} \rightarrow \mathbb {R}$, defined by

$\varphi (x)=x^{4}+3x^{2}-2x+5\,\,{\text{ and }}\,\,\psi (x)=2x^{3}-x^{2}+6x-1.$

Exercise Create referencenumber

1. Can a constant mapping be bijective?
2. Is the composition of a constant mapping with an arbitrary mapping (the constant map first) constant?
3. Is the composition of an arbitrary mapping with a constant mapping (the constant map last) constant?

===Exercise * Exercise 3.27

change===

Let ${}L,M,N$ and ${}P$ be sets and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

$G\colon M\longrightarrow N,y\longmapsto G(y),$

and

$H\colon N\longrightarrow P,z\longmapsto H(z),$

be functions.MDLD/functions Show that

${}H\circ (G\circ F)=(H\circ G)\circ F\,.$

Exercise Create referencenumber

Let ${}L,\,M$ and ${}N$ denote sets and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappingsMDLD/mappings with the compositionMDLD/composition

$G\circ F\colon L\longrightarrow N.$

Show the following properties.

1. If ${}F$ and ${}G$ are injective,MDLD/injective then also ${}G\circ F$ is injective.
2. If ${}F$ and ${}G$ are surjective,MDLD/surjective then also ${}G\circ F$ is surjective.
3. If ${}F$ and ${}G$ are bijective,MDLD/bijective then also ${}G\circ F$ is bijective.

Exercise * Create referencenumber

Let ${}L,M,N$ be sets and let

$f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$

be mappingsMDLD/mappings with their compositionMDLD/composition

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$

Show that if ${}g\circ f$ is injective,MDLD/injective then also ${}f$ is injective.

Hand-in-exercises

Exercise (4 marks) Create referencenumber

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 (2 marks) Create referencenumber

Sketch the following subsets in ${}\mathbb {R} ^{2}$.

1. ${}{\left\{(x,y)\mid 2x=5{\text{ and }}y\geq 3\right\}}$,
2. ${}{\left\{(x,y)\mid -3x\geq 2y{\text{ and }}4x\leq -5y\right\}}$,
3. ${}{\left\{(x,y)\mid y^{2}-y+1\leq 4\right\}}$,
4. ${}{\left\{(x,y)\mid xy=0\right\}}$.

Exercise (3 marks) Create referencenumber

Let ${}L,M,N$ be sets, and let

$f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$

be mappingsMDLD/mappings with their composite mappingMDLD/composite mapping

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$

Show that if ${}g\circ f$ is surjective,MDLD/surjective then also ${}g$ is surjective.

Exercise (4 marks) Create referencenumber

We consider a computer with only two memory units, each can represent a natural number. At the start of every program (a sequence of commands), the initial entries are ${}(0,0)$. The computer can empty a memory unit, it can increase a memory unit by ${}1$, it can jump to another command (unconditional Goto) and it can compare the two entries of the two memory units. Moreover, it can jump to another command in case the comparing condition is fulfilled (conditional Goto). There is a printing command which prints the entries at the moment. Write a computer-program such that every pair ${}(n,m)\in \mathbb {N} ^{2}$ is printed exactly once.

Exercise (4 marks) Create referencenumber

Determine the compositionsMDLD/compositions ${}\varphi \circ \psi$ and ${}\psi \circ \varphi$ for the mappingsMDLD/mappings ${}\varphi ,\psi \colon \mathbb {R} \rightarrow \mathbb {R}$ given by

$\varphi (x)=x^{3}+2x+1\,\,{\text{ and }}\,\,\psi (x)=x^{2}-5.$

Exercise (3 marks) Create referencenumber

Consider the set ${}M=\{1,2,3,4,5,6,7,8\}$, and the mappingMDLD/mapping

$\varphi \colon M\longrightarrow M,x\longmapsto \varphi (x),$

defined by the following table

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$	${}8$
${}\varphi (x)$	${}2$	${}5$	${}6$	${}1$	${}4$	${}3$	${}7$	${}7$

Compute ${}\varphi ^{1003}$, that is, the ${}1003$-rd composition (or iteration) of ${}\varphi$ with itself.

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