discrete_mathematics_with_applications/chapter_6/exercises.md

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Exercise Set 6.1

1. In each of (a) -(f), answer the following questions: Is A \subseteq B? Is B \subseteq A? Is either A or B a proper subset of the other?

a. A = \{2, \{2\}, (\sqrt{2})^2\}, B = \{2, \{2\}, \{\{2\}\}\}

b. A = \{3, \sqrt{5^2 - 4^2}, 24 \mod 7\}, B = \{8 \mod 5\}

c. A = \{\{1, 2\}, \{2, 3\}\}, B = \{1, 2, 3\}

d. A = \{a, b, c\}, B = \{\{a\}, \{b\}, \{c\}\}

e. A = \{\sqrt{16}, \{4\}\}, B = \{4\}

f. A = \{x \in \mathbb{R} | \cos x \in \mathbb{Z}\}, B = \{x \in \mathbb{R} | \sin x \in \mathbb{Z}\}

2. Complete the proof from Example 6.1.3: Prove that B \subseteq A where

 A = \{m \in \mathbb{Z} | m, = 2a \text{ for some integer } a\} 

and

 B = \{n \in \mathbb{Z} | n = 2b - 2 \text{ for some integer } b\} 

3. Let sets R, S, and T be defined as follows:

 R = \{x \in \mathbb{Z} | x \text{ is divisible by } 2\}  

 S = \{y \in \mathbb{Z} | y \text{ is divisible by } 3\}  

 T = \{z \in \mathbb{Z} | z \text{ is divisible by } 6\}  

Prove or disprove each of the following statements.

a. R \subseteq T

b. T \subseteq R

c. T \subseteq S

4. Let A = \{n \in \mathbb{Z} | n = 5r \text{ for some integer } r\} and B = \{m \in \mathbb{Z} | m = 20s \text{ for some integer } s\}. Prove or disprove each of the following statements.

a. A \subseteq B

b. B \subseteq A

5. Let C = \{n \in \mathbb{Z} | n = 6r - 5 \text{ for some integer } r\} and D = \{m \in \mathbb{Z} | m = 3s + 1 \text{ for some integer } s\}. Prove or disprove each of the following statements.

a. C \subseteq D

b. D \subseteq C

6. Let A = \{x \in \mathbb{Z} | x = 5a + 2 \text{ for some integer } a\}, B = \{y \in \mathbb{Z} | y = 10b - 3 \text{ for some integer } b\}, and C = \{z \in \mathbb{Z} | z = 10c + 7 \text{ for some integer } c\}.

Prove or disprove each of the following statements.

a. A \subseteq B

b. B \subseteq A

c. B = C

7. Let A = \{x \in \mathbb{Z} | x = 6a + 4 \text{ for some integer } a\}, B = \{y \in \mathbb{Z} | y = 18b - 2 \text{ for some integer } b\}, and C = \{z \in \mathbb{Z} | z = 18c + 16 \text{ for some integer } c\}.

Prove or disprove each of the following statements.

a. A \subseteq B

b. B \subseteq A

c. B = C

8. Write in words to read each of the following out loud. Then write each set using the symbols for union, intersection, set difference, or set complement.

a. \{x \in U | x \in A \text{ and } x \in B\}

b. \{x \in U | x \in A \text{ or } x \in B\}

c. \{x \in U | x \in A \text{ and } x \notin B\}

d. \{x \in U | x \notin A\}

9. Complete the following sentences without using the symbols \cup, \cap, or -.

a. x \notin A \cup B if, and only if, _____.

b. x \notin A \cap B if, and only if, _____.

c. x \notin A - B if, and only if, _____.

10. Let A = \{1, 3, 5, 7, 9\}, b = \{3, 6, 9\}, and C = \{2, 4, 6, 8\}. Find each of the following:

a. A \cup B

b. A \cap B

c. A \cup C

d. A \cap C

e. A - B

f. B - A

g. B \cup C

h. B \cap C

11. Let the universal set \mathbb{R}, the set of all real numbers, and let A = \{x \in \mathbb{R} | 0 < x \leq 2\}, B = \{x \in \mathbb{R} | 1 \leq x < 4\}, and C = \{x \in \mathbb{R} | 3 \leq x < 9\}. Find each of the following:

a. A \cup B

b. A \cap B

c. A^c

d. A \cup C

e. A \cap C

f. B^c

g. A^c \cap B^c

h. A^c \cup B^c

i. (A \cap B)^c

j. (A \cup B)^c

12. Let the universal set be \mathbb{R}, the set of all real numbers, and let A = \{x \in \mathbb{R} | -3 \leq x \leq 0\}, B = \{x \in \mathbb{R} | -1 < x < 2\}, and C = \{x \in \mathbb{R} | 6 < x \leq 8\}. Find each of the following:

a. A \cup B

b. A \cap B

c. A^c

d. A \cup C

e. A \cap C

f. B^c

g. A^c \cap B^c

h. A^c \cup B^c

i. (A \cap B)^c

j. (A \cup B)^c

13. Let S be the set of all strings of $0$'s and $1$'s of length 4, and let A and B be the following subsets of S: A = \{1110, 1111, 1000, 1001\} and B = \{1100, 0100, 1111, 0111\}. Find each of the following:

a. A \cap B

b. A \cup B

c. A - B

d. B - A

14. In each of the following, draw a Venn diagram for sets A, B, and C that satisfy the given conditions.

a. A \subseteq B, C \subseteq B, A \cap C = \emptyset

b. C \subseteq A, B \cap C = \emptyset

15. In each of the following, draw a Venn diagram for sets A, B, and C that satisfy the given conditions.

a. A \cap B = \emptyset, A \subseteq C, C \cap B \neq \emptyset

b. A \subseteq B, C \subseteq B, A \cap C \neq \emptyset

c. A \cap B \neq \emptyset, B \cap C \neq \emptyset, A \cap C = \emptyset, A \nsubseteq B, C \nsubseteq B

16. Let A = \{a, b, c\}, B = \{b, c, d\}, and C = \{b, c, e\}.

a. Find A \cup (B \cap C), (A \cup B) \cap C, and (A \cup B) \cap (A \cup C). Which of these sets are equal?

b. Find A \cap (B \cup C), (A \cap B) \cup C, and (A \cap B) \cup (A \cap C). Which of these sets are equal?

c. Find (A - B) - C and A - (B - C). Are these sets equal?

17. Consider the following Venn diagram. For each of (a)-(f), copy the diagram and shade the region corresponding to the indicated set.

a. A \cap B

b. B \cup C

c. A^c

d. A - (B \cup C)

e. (A \cup B)^c

f. A^c \cap B^c

(See page 412 for image)

a. Is the number 0 in \emptyset? Why?

b. Is \emptyset = \{\emptyset\}? Why?

c. Is \emptyset \in \{\emptyset\} Why?

d. Is \emptyset \in \emptyset? Why?

19. Let A_i = \{i, i^2\} for each integer i = 1, 2, 3, 4.

a. A_1 \cup A_2 \cup A_3 \cup A_4 = \text{ ?}

b. A_1 \cap A_2 \cap A_3 \cap A_4 = \text{ ?}

c. Are A_1, A_2, A_3, and A_4 mutually disjoint? Explain.

20. Let B_i = \{x \in \mathbb{R} | 0 \leq x\leq i\} for each integer i = 1, 2, 3, 4.

a. B_1 \cup B_2 \cup B_3 \cup B_4 = \text{ ?}

b. B_1 \cap B_2 \cap B_3 \cap B_4 = \text{ ?}

c. Are B_1, B_2, B_3, and B_4 mutually disjoint? Explain.

21. Let C_i = \{i, -i\} for each nonnegative integer i.

a. \bigcup_{i = 0}^{4}C_i = \text{ ?}

b. \bigcap_{i = 0}^{4}C_i = \text{ ?}

c. Are C_0, C_1, C_2, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}C_i = \text{ ?}

e. \bigcap_{i = 0}^{n}C_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty}C_i = \text{ ?}

g. \bigcap_{i = 0}^{\infty}C_i = \text{ ?}

22. Let D_i = \{x \in \mathbb{R} | -i \leq x \leq i\} = [-i, i] for each nonnegative integer i.

a. \bigcup_{i = 0}^{4}D_i = \text{ ?}

b. \bigcap_{i = 0}^{4}D_i = \text{ ?}

c. Are D_0, D_1, D_2, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}D_i = \text{ ?}

e. \bigcap_{i = 0}^{n}D_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty}D_i = \text{ ?}

g. \bigcap_{i = 0}^{\infty}D_i = \text{ ?}

23. Let V_i = \{x \in \mathbb{R} | -\dfrac{1}{i} \leq x \leq \dfrac{1}{i}\} = \left[-\dfrac{1}{i}, \dfrac{1}{i}\right] for each positive integer i.

a. \bigcup_{i = 0}^{4}V_i = \text{ ?}

b. \bigcap_{i = 0}^{4}V_i = \text{ ?}

c. Are V_1, V_2, V_3, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}V_i = \text{ ?}

e. \bigcap_{i = 0}^{n}V_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty} = \text{ ?}

g. \bigcap_{i = 0}^{\infty} = \text{ ?}

24. Let W_i = \{x \in \mathbb{R} | x > i\} = (i, \infty) for each nonnegative integer i.

a. \bigcup_{i = 0}^{4}W_i = \text{ ?}

b. \bigcap_{i = 0}^{4}W_i = \text{ ?}

c. Are W_0, W_1, W_2, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}W_i = \text{ ?}

e. \bigcap_{i = 0}^{n}W_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty}W_i = \text{ ?}

g. \bigcap_{i = 0}^{\infty}W_i = \text{ ?}

25. Let R_i = \{x \in \mathbb{R} | 1 \leq x \leq 1 + \dfrac{1}{i}\} = \left[1, 1 + \dfrac{1}{i}\right] for each positive integer i.

a. \bigcup_{i = 0}^{4}R_i = \text{ ?}

b. \bigcap_{i = 0}^{4}R_i = \text{ ?}

c. Are R_1, R_2, R_3, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}R_i = \text{ ?}

e. \bigcap_{i = 0}^{n}R_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty}R_i = \text{ ?}

g. \bigcap_{i = 0}^{\infty}R_i = \text{ ?}

26. Let S_i = \{x \in \mathbb{R} | 1 < x < 1 + \dfrac{1}{i}\} = \left(1, 1 + \dfrac{1}{i}\right) for each positive integer i.

a. \bigcup_{i = 0}^{4}S_i = \text{ ?}

b. \bigcap_{i = 0}^{4}S_i = \text{ ?}

c. Are S_1, S_2, S_3, \dots mutually disjoint? Explain.

d. \bigcup_{i = 0}^{n}S_i = \text{ ?}

e. \bigcap_{i = 0}^{n}S_i = \text{ ?}

f. \bigcup_{i = 0}^{\infty}S_i = \text{ ?}

g. \bigcap_{i = 0}^{\infty}S_i = \text{ ?}

a. Is \{\{a, d, e\}, \{b, c\}, \{d, f\}\} a partition of \{a, b, c, d, e, f\}?

b. Is \{\{w, x, v\}, \{u, y, q\}, \{p, z\}\} a partition of \{p, q, u, v, w, x, y, z\}?

c. Is \{\{5, 4\}, \{7, 2\}, \{1, 3, 4\}, \{6, 8\}\} a partition of \{1, 2, 3, 4, 5, 6, 7, 8\}?

d. Is \{\{3, 7, 8\}, \{2, 9\}, \{1, 4, 5\}\} a partition of \{1, 2, 3, 4, 5, 6, 7, 8, 9\}?

e. Is \{\{1, 5\}, \{4, 7\}, \{2, 8, 6, 3\}\} a partition of \{1, 2, 3, 4, 5, 6, 7, 8\}?

28. Let E be the set of all even integers and O the set of all odd integers. Is \{E, O\} a partition of \mathbb{Z}, the set of all integers? Explain your answer.

29. Let \mathbb{R} be the set of all real numbers. Is \{\mathbb{R}^+, \mathbb{R}^-, \{0\}\} a partition of \mathbb{R}? Explain your answer.

30. Let \mathbb{Z} be the set of all integers and let

 A_0 = \{n \in \mathbb{Z} | n = 4k, \text{ for some integer } k\} 

 A_1 = \{n \in \mathbb{Z} | n = 4k + 1, \text{ for some integer } k\} 

 A_2 = \{n \in \mathbb{Z} | n = 4k + 2, \text{ for some integer } k\} 

and

 A_3 = \{n \in \mathbb{Z} | n = 4k + 3, \text{ for some integer } k\} 

Is \{A_0, A_1, A_2, A_3\} a partition of \mathbb{Z}? Explain your answer.

31. Suppose A = \{1, 2\} and B = \{2, 3\}. Find each of the following:

a. \mathscr{P}(A \cap B)

b. \mathscr{P}(A)

c. \mathscr{P}(A \cup B)

d. \mathscr{P}(A \times B)

a. Suppose A = \{1\} and B = \{u, v\}. Find \mathscr{P}(A \times B).

b. Suppose X = \{a, b\} and Y = \{x, y\}. Find \mathscr{P}(X \times Y).

a. Find \mathscr{P}(\emptyset).

b. Find \mathscr{P}(\mathscr{P}(\emptyset)).

b. Find \mathscr{P}(\mathscr{P}(\mathscr{P}(\emptyset))).

34. let A_1 = \{1\}, A_2 = \{u, v\}, and A_3 = \{m, n\}. Find each of the following sets:

a. A_1 \cup (A_2 \times A_3)

b. (A_1 \cup A_2) \times A_3

35. let A = \{a, b\}, B = \{1, 2\}, and C = \{2, 3\}. Find each of the following sets:

a. A \times (B \cup C)

b. (A \times B) \cup (A \times C)

c. A \times (B \cap C)

d. (A \times B) \cap (A \times C)

36. Trace the action of Algorithm 6.1.1 on the variables i, j, \text{found}, and \text{answer} for m = 3, n = 3, and sets A and B represented as the arrays a[1] = u, a[2] = v, a[3] = w, b[1] = w, b[2] = u, and b[3] = v.

37. Trace the action of Algorithm 6.1.1 on the variables i, j, \text{found}, and \text{answer} for m = 4, n = 4 and sets A and B represented as the arrays a[1] = u, a[2] = v, a[3] = w, a[4] = x, b[1] = r, b[2] = u, b[3] = y, b[4] = z.

38. Write an algorithm to determine whether a given element x belongs to a given set that is represented as the array a[1], a[2], \dots, a[n].