Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 1

Exercise for the break

Exercise

Draw, for four sets, a diagram of sets that shows all possible intersecting sets.

An abstract and

Exercises

Exercise

a concrete set diagram.

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. $(LA\cap GA)\cup (LA\cap RA)$.
3. $RA\setminus (GA\cup RA)$.
4. $RA\setminus (GA\cup LA)$.
5. $(RA\setminus GA)\cap ((LA\cup GA)\setminus (GA\cap RA))$.

Exercise

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. $(N\cup P)\cap R$,
5. $N\setminus R$,
6. $(M\cup P)\setminus (R\setminus N)$,
7. $((P\cup R)\cap N)\cap R$,
8. $(R\setminus P)\cap (M\setminus N)$.

Exercise

Sketch the following subsets of ${ℝ}^2$.

1. $\{(x,y)\mid x=5\}$,
2. $\{(x,y)\mid x\ge 4\text{ and }y=3\}$,
3. $\{(x,y)\mid y^2\ge 2\}$,
4. $\{(x,y)\mid |x|=3\text{ and }|y|\le 2\}$,
5. $\{(x,y)\mid 3x\ge y\text{ and }5x\le 2y\}$,
6. $\{(x,y)\mid xy=0\}$,
7. $\{(x,y)\mid xy=1\}$,
8. $\{(x,y)\mid xy\ge 1\text{ and }y\ge x^3\}$,
9. $\{(x,y)\mid 0=0\}$,
10. $\{(x,y)\mid 0=1\}$.

Exercise

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

$A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C).$

Exercise

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

1. $A\cup \mathrm{\varnothing }=A,$

2. $A\cap \mathrm{\varnothing }=\mathrm{\varnothing },$

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

Let $M$ and $N$ be disjoint sets, and $x\in M$ Show that also $M\setminus \{x\}$ and $N\cup \{x\}$ are disjoint, and that

$M\cup N=(M\setminus \{x\})\cup (N\cup \{x\})$

holds.

Exercise

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

Exercise

We consider the two sets

$E=\{\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\in {ℝ}^3\mid -3x+2y-6z=0\}$

and

$F=\{\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\in {ℝ}^3\mid 7x-5y-4z=0\}.$

Find a description of the intersection

$G:=E\cap F=\{\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\in {ℝ}^3\mid -3x+2y-6z=0\text{ and }7x-5y-4z=0\},$

similar to Example 1.2 .

Exercise *

1. Show that the set

   $\{(x,y)\in {\mathbb{Z}}^2\mid 3x+5y=1\}$

   is not empty.

2. Show that the set

   $\{(x,y)\in {\mathbb{Z}}^2\mid 6x+9y=5\}$

   is empty.

Exercise

On the dating-site "Catch your match“, there is a set $M$ of registered users. Moreover, there is a set $E$ of properties, which the users satisfy or not (this can be seen in their profiles). For a subset of properties $W\subseteq E$ (wanted properties), we define

$W^{\perp }=\{m\in M\mid m\text{ fulfills all properties from }W\},$

and for a subset $S\subseteq M$, we define

$S^{\perp }=\{e\in E\mid \text{ every person from }S\text{ fulfills the property }e\}.$

1. Describe, for a property $e\in E$, the set $\{e{\}}^{\perp }$ with a sentence.
2. Describe, for a person $m\in M$, the set $\{m{\}}^{\perp }$ with a sentence.
3. Why do we (probably) have $E^{\perp }=\mathrm{\varnothing }$?
4. Show: for subsets $W_1\subseteq W_2$ (in $E$), we have

   ${\left(W_1\right)}^{\perp }\supseteq {\left(W_2\right)}^{\perp }.$

5. Show: for an arbitrary subset $W\subseteq E$, we have

   $W\subseteq {\left(W^{\perp }\right)}^{\perp }.$

6. Show: for a union

   $W=W_1\cup W_2\subseteq E,$

   we have

   ${(W_1\cup W_2)}^{\perp }={\left(W_1\right)}^{\perp }\cap {\left(W_2\right)}^{\perp }.$

7. Does, for an intersection

   $W=W_1\cap W_2\subseteq E,$

   the relation

   ${(W_1\cap W_2)}^{\perp }={\left(W_1\right)}^{\perp }\cup {\left(W_2\right)}^{\perp }$

   hold?

8. Does, for an arbitrary subset $W\subseteq E$, the relation

   $W^{\perp }={\left({\left(W^{\perp }\right)}^{\perp }\right)}^{\perp }$

   hold?

How can we describe the round bale of straw (without the cat) as a product set?

Exercise

Describe, for every combination (including the case where the product is taken with itself), the product set of the following geometric sets.

1. A circle $K$.
2. A line segment $I$.
3. A line $G$.
4. A parabola $P$.

Which product set can we realize as a subset in space?

Exercise

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

$(A\times N)\cap (M\times B)=A\times B.$

Exercise

Let $A$ and $B$ denote disjoint sets, and let $C$ be another set. Show the identity

$C\times (A\uplus B)=(C\times A)\uplus (C\times B).$

Exercise

Let $A$ and $B$ denote disjoint sets. Show the equality

$(A\uplus B)\times (A\uplus B)=(A\times A)\uplus (A\times B)\uplus (B\times A)\uplus (B\times B).$

Exercise

Let $G$ denote a finite set with $n$ elements. Show that the power set $𝔓(G)$ contains $2^n$ elements.

Hand-in-exercises

Exercise (2 marks)

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

1. $\{(x,y)\mid |2x|=5\text{ and }|y|\ge 3\}$,
2. $\{(x,y)\mid -3x\ge 2y\text{ and }4x\le -5y\}$,
3. $\{(x,y)\mid y^2-y+1\le 4\}$,
4. $\{(x,y)\mid xy=2\text{ or }{x}^2+y^2=1\}$.

Exercise (2 (1+1) marks)

1. Sketch the set $M=\{(x,y)\in {ℝ}^2\mid -5x+2y=6\}$ and the set $N=\{(x,y)\in {ℝ}^2\mid 7x-5y=4\}$.
2. Determine the intersection $M\cap N$ by a drawing and computationally.

Exercise (1 mark)

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 (5 marks)

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: $B\cap A=\mathrm{\varnothing }$ and $C\subseteq B$ imply $C\cap A=\mathrm{\varnothing }$.
3. Modus Darii: $B\subseteq A$ and $C\cap B\ne \mathrm{\varnothing }$ imply $C\cap A\ne \mathrm{\varnothing }$.
4. Modus Ferio: $B\cap A=\mathrm{\varnothing }$ and $C\cap B\ne \mathrm{\varnothing }$ imply $C\subseteq ̸A$.
5. Modus Baroco: $B\subseteq A$ and $B\subseteq ̸C$ imply $A\subseteq ̸C$.

Exercise (2 marks)

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

$(A_1\times B_1)\cap (A_2\times B_2)=(A_1\cap A_2)\times (B_1\cap B_2).$

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. $A\setminus B=\mathrm{\varnothing }$,
5. There exists a set $C$ such that $B=A\cup C$,
6. There exists a set $D$ such that $A=B\cap D$.

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