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 3.2.6 =

= Exercise 3.2.7 =

= Exercise 3.2.8 =

= 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 &isin; A, then x &isin; B, and because x &isin; B, x &isin; C. The same goes for any y &isin; B or any z &isin; C.

= Exercise 3.2.11 =

= 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 3.2.14 =

= Exercise 3.2.15 =

1
$$\mathcal P( \mathcal P ( \emptyset )) = \{ \emptyset, \{ \emptyset \} \} $$

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

= Exercise 3.2.16 = (1) false (2) true (3) true (4) true (5) false (6) true (7) false (8) false (9) true