Book of Proof - Amazon S3

26 Sets
Example 1.11 Here the sets A α will be subsets of R 2 . Let I = [0,2] =
{
x ∈ R : 0 ≤ x ≤ 2
} . For each number α ∈ I, let Aα = { (x,α) : x ∈ R,1 ≤ x ≤ 2 } . For
instance, given α = 1 ∈ I the set A 1 = { (x,1) : x ∈ R,1 ≤ x ≤ 2 } is a horizontal
line segment one unit above the x-axis and stretching between x = 1 and
x = 2, as shown in Figure 1.11(a). Likewise A 2 = { (x, 2) : x ∈ R,1 ≤ x ≤ 2 } is
a horizontal line segment 2 units above the x-axis and stretching between
x = 1 and x = 2. A few other of the A α are shown in Figure 1.11(a), but they
can’t all be drawn because there is one A α for each of the infinitely many
numbers α ∈ [0,2]. The totality of them covers the shaded region in Figure
1.11(b), so this region is the union of all the A α . Since the shaded region
is the set { (x, y) ∈ R 2 : 1 ≤ x ≤ 2,0 ≤ y ≤ 2 } = [1,2] × [0,2], it follows that
⋃
α∈[0,2]
A α = [1,2] × [0,2].
Likewise, since there is no point (x, y) that is in every set A α , we have
⋂
α∈[0,2]
A α = .
y
y
2
A 2
2
1
A 2
A 1
1
⋃
A α
α∈[0,2]
1
A 0.5
A 0.25
x
2
1
2
x
(a)
(b)
Figure 1.11. The union of an indexed collection of sets
One final comment. Observe that A α = [1,2] × { α } , so the above expressions
can be written as
⋃
[1,2] × { α } ⋂
= [1,2] × [0,2] and [1,2] × { α } = .
α∈[0,2]
α∈[0,2]