# Exercise 3.2.1

3, namely $a,b$ and $\{a,b\}$ 1. False

2. True

3. True

4. True

5. False

6. False

7. False

8. True

9. True

# Exercise 3.2.3

## 1

The set of even integers

## 2

The set of composite numbers

## 3

The set of all rational numbers.

# Exercise 3.2.4

## 1

The set of all fathers

## 2

The set of all grandparents

## 3

The set of all people that are married to a woman

## 4

The set of all siblings

## 5

The set of all people that are younger than someone

## 6

The set of all people that are older than their father

# Exercise 3.2.5

## 1

$\{x\in R|x>0\}$ ## 2

$\{x\in Z|$ there exist $y\in Z$ such that $x=2*y+1\}$ ## 3

$\{x\in R|$ there exist $y\in N$ such that $5*y*x=1\}$ ## 5

$\{x\in N|$ there exist $y\in N$ such that $x=4*y+1\}$ # Exercise 3.2.9

A = {1,2}, B = {1,2,{1,2}}

# Exercise 3.2.10

Using the definition of a subset: For any xA, then xB, and because xB, xC. The same goes for any yB or any zC.

# Exercise 3.2.12

False. Counterexample. Let A be a set of even integers and B a set of odd integers.Then A and B are not equal, and A is not a subset of B, and B is not a subset of A. A and B are disjoint.

# Exercise 3.2.13

${\mathcal {P}}(A)=\{\emptyset ,x,y,z,w,\{x,y\},\{x,z\},\{x,w\},\{y,z\},\{y,w\},\{z,w\},\{x,y,z\},\{x,y,w\},\{x,z,w\},\{y,z,w\},\{x,y,z,w\}\}$ # Exercise 3.2.15

## 1

${\mathcal {P}}({\mathcal {P}}(\emptyset )=\{\emptyset ,\{\emptyset \}\}$ ## 2

[itex]\mathcal P( \mathcal P ( \{ \emptyset \} ) = \{ \emptyset, \{ \emptyset \}, \{ \{ \

# Exercise 3.2.16

(1) false (2) true (3) true (4) true (5) false (6) true (7) false (8) false (9) true