Answers to Exer cises
CHAPTER 0
1. (a) No (b) No (c) Yes
2. (a) Yes (b) No (c) Yes
3. (a) True (b) True (c) False
4. Suppose that x A
∈ . Since A B
⊆ , this implies that x B
∈ . Since B C
⊆ ,
we have x C
∈ . Since x A
∈ implies that x C
∈ , it follows that A C
⊆ .
5. (a) 1 (b) 1 (c) 2 (d) 3
6. (a) { , { }}
φ a
(b) { , { }, { }, { , }}
φ a b a b
(c) { , { }, {{ }}, { , { }}}
φ φ
φ
φ φ
7. (a) 2 (b) 16
8. (a) {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )}
a y b y c y d y a z b z c z d z
(b) {( , ), ( , ), ( , )( , ), ( , ), ( , ), ( , ), ( , )}
y a y b y c y d z a z b z c z d
9. The set of triples (a, b, c) where a is an airline and b and c are cities.
10. φ
φ
φ
φ
φ
× =
∈
∈
= =
∈
∈
= ×
Α {( , ) |
}
{( , |
}
x y x
y A
x y x A
y
A
and
and
11. mn
13. (a) {0, 1, 2, 3, 4, 5, 6}
(b) {3}
(c) {1, 2, 4, 5}
(d) {0,6}
18. (a) B A
⊆
(b) A B
⊆
(c) A B
∩ = φ
(d) Nothing, since this is always true
(e) A = B
19. (a) { , , ,
}
1 2 3 KK n
(b) {1}