Texts: Please help with questions 2, 4, and 6. Thank you very much.
EXERCISE SET 6.1*
1. In each of (a) to (f), answer the following questions: Is A ⊆ B? Is B ⊆ A? Is either A or B a proper subset of the other?
a. A = {2, {2}, √22}, B = {2, {2}, {{2}}}
b. A = {3, √52-42, 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 = {√16, {4}}, B = {4}
f. A = {x ∈ ℝ | cos x ∈ ℤ}, B = {x ∈ ℝ | sin x ∈ ℤ}
5. Let C = {n ∈ ℤ | n = 6r + 5 for some integer r} and D = {m ∈ ℤ | m = 3s + 1 for some integer s}. Prove or disprove each of the following statements.
a. C ⊆ D
b. D ⊆ C
6. Let A = {x ∈ ℤ | x = 5a + 2 for some integer a}, B = {y ∈ ℤ | y = 10b - 3 for some integer b}, and C = {z ∈ ℤ | z = 10c + 7 for some integer c}. Prove or disprove each of the following statements.
a. A ⊆ B
b. B ⊆ C
c. B = C
2. Complete the proof from Example 6.1.3: Prove that B ⊆ A where...
7. Let A = {x ∈ ℤ | x = 6a + 4 for some integer a}, B = {y ∈ ℤ | y = 18b + 2 for some integer b}, and C = {z ∈ ℤ | z = 18c + 16 for some integer c}.
A = {m ∈ ℤ | m = 2a for some integer a}
Prove or disprove each of the following statements.
a. A ⊆ B
b. B ⊆ A
c. B = C
8. Write in words how 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 ∈ ℤ | x ∈ A and x ∈ B}
b. {x ∈ ℤ | x ∈ A or x ∈ B}
c. {x ∈ ℤ | x ∈ A and x ∈ B}
d. {x ∈ ℤ | x ∈ A}
9. Complete the following sentences without using the symbols U, ∩, or ':
a. A ∪ B if, and only if...
b. A ∩ B if, and only if...
c. A - B if, and only if...
B = {n ∈ ℤ | n = 2b - 2 for some integer b}
3. Let sets R, S, and T be defined as follows:
R = {x ∈ ℤ | x is divisible by 2}
S = {y ∈ ℤ | y is divisible by 3}
T = {z ∈ ℤ | z is divisible by 6}.
Prove or disprove each of the following statements.
a. R ⊆ T
b. T ⊆ R
c. T ⊆ S
4. Let A = {n ∈ ℤ | n = 5r for some integer r} and B = {m ∈ ℤ | m = 20s for some integer s}. Prove or disprove each of the following statements.
a. A ⊆ B
b. B ⊆ A
*For exercises with blue numbers or letters, solutions are given in Appendix B. The symbol H indicates that only a hint or a partial solution is given. The symbol signals that an exercise is more challenging than usual.
