Introduction to Abstract Algebra, Solutions Manual, 4th Edition

0.2 Sets

1.

a. A = {x x = 5k, , k ≥ 1}

2.

a. {1, 3, 5, 7, . . . }

c. { − 1, 1, 3}

e. { } = ∅ is the empty set by Example 3.

3.

a. Not equal: −1 A but −1 ∉ B.

c. Equal to {a, l, o, y}.

e. Not equal: 0 A but 0 ∉ B.

g. Equal to { − 1, 0, 1}.

4.

a. ∅, {2}

c. {1}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

5.

a. True. B ⊆ C means each element of B (in particular A) is an element of C.

c. False. For example, A = {1}, B = C = {{1}, 2}.

6.

a. Clearly A ∩ B ⊆ A and A ∩ B ⊆ B; If X ⊆ A and X ⊆ B, then x X implies x A and x B, that is x A ∩ B. Thus X ⊆ A ∩ B.

7. If x A ∪ (B₁ ∩ B₂ ∩ ...