Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 3

Exercise 3.2.1

3, namely $a,b$ and $\{a,b\}$

Exercise 3.2.2

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 $5*y*x=1\}$

4

{n^3|n is an integer and -5<n<5}

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 x ∈ A, then x ∈ B, and because x ∈ B, x ∈ C. The same goes for any y ∈ B or any z ∈ C.

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

$𝒫(A)=\{\mathrm{\varnothing },x,y,z,w,\{x,y\},\{x,z\},\{x,w\},\{y,z\},\{y,w\},\{z,w\},\{x,y,z\},\{x,y,w\},\{x,z,w\},\{y,z,w\},\{x,y,z,w\}\}$

Exercise 3.2.15

1

$𝒫(𝒫(\mathrm{\varnothing }))=\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$

2

$𝒫(𝒫(\{\mathrm{\varnothing }\}))=\{\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\},\{\mathrm{\varnothing }\},\{\{\mathrm{\varnothing }\}\},\mathrm{\varnothing }\}$

Exercise 3.2.16

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