Book of Proof - Amazon S3

Indexed Sets 27
Exercises for Section 1.8
1. Suppose A 1 = { a, b, d, e, g, f } , A 2 = { a, b, c, d } , A 3 = { b, d, a } and A 4 = { a, b, h } .
4⋃
4⋂
(a) A i =
(b) A i =
i=1
⎧
⎨ A 1 = { 0,2,4,8,10,12,14,16,18,20,22,24 } ,
2. Suppose A
⎩ 2 = { 0,3,6,9,12,15,18,21,24 } ,
A 3 = { 0,4,8,12,16,20,24 } .
3⋃
3⋂
(a) A i =
(b) A i =
i=1
i=1
i=1
3. For each n ∈ N, let A n = { 0,1,2,3,..., n } .
⋃
(a) A i =
i∈N
(b)
⋂
A i =
i∈N
4. For each n ∈ N, let A n = { − 2n,0,2n } .
⋃
(a) A i =
5. (a)
6. (a)
7. (a)
8. (a)
9. (a)
10. (a)
i∈N
⋃
[i, i + 1] =
i∈N
(b)
(b)
⋃
[0, i + 1] = (b)
i∈N
⋃
R × [i, i + 1] =
i∈N
(b)
⋂
A i =
i∈N
⋂
[i, i + 1] =
i∈N
⋂
[0, i + 1] =
i∈N
⋂
R × [i, i + 1] =
⋃ { } ⋂ { }
α × [0,1] = (b) α × [0,1] =
α∈R
⋃
X∈P(N)
⋃
x∈[0,1]
X =
[x,1] × [0, x 2 ] =
(b)
(b)
i∈N
α∈R
⋂
X∈P(N)
⋂
x∈[0,1]
X =
[x,1] × [0, x 2 ] =
11. Is ⋂ A α ⊆ ⋃ A α always true for any collection of sets A α with index set I?
α∈I α∈I
12. If ⋂ A α = ⋃ A α , what do you think can be said about the relationships between
α∈I α∈I
the sets A α ?
13. If J ≠ and J ⊆ I, does it follow that ⋃ A α ⊆ ⋃ A α ? What about ⋂ A α ⊆ ⋂ A α ?
α∈J α∈I
α∈J α∈I
14. If J ≠ and J ⊆ I, does it follow that ⋂ A α ⊆ ⋂ A α ? Explain.
α∈I α∈J