Book of Proof - Amazon S3

Indexed Sets 25
This notation is also used when the list of sets A 1 , A 2 , A 3 , ... is infinite:
∞⋃
A i = A 1 ∪ A 2 ∪ A 3 ∪ ··· = { x : x ∈ A i for at least one set A i with 1 ≤ i } .
i=1
∞⋂
A i = A 1 ∩ A 2 ∩ A 3 ∩ ··· = { x : x ∈ A i for every set A i with 1 ≤ i } .
i=1
Example 1.10
This example involves the following infinite list of sets.
A 1 = { − 1,0,1 } , A 2 = { − 2,0,2 } , A 3 = { − 3,0,3 } , ··· , A i = { − i,0, i } , ···
⋃
Observe that ∞ ⋂
A i = Z, and ∞ A i = { 0 } .
i=1
i=1
Here is a useful twist on our new notation. We can write
3⋃
A i =
⋃
A i ,
i=1 i∈{1,2,3}
as this takes the union of the sets A i for i = 1,2,3. Likewise:
3⋂
A i =
i=1
∞⋃
A i =
i=1
∞⋂
A i =
i=1
⋂
A i
i∈{1,2,3}
⋃
A i
i∈N
⋂
A i
i∈N
Here we are taking the union or intersection of a collection of sets A i
where i is an element of some set, be it { 1,2,3 } or N. In general, the way
this works is that we will have a collection of sets A i for i ∈ I, where I is
the set of possible subscripts. The set I is called an index set.
It is important to realize that the set I need not even consist of integers.
(We could subscript with letters or real numbers, etc.) Since we are
programmed to think of i as an integer, let’s make a slight notational
change: We use α, not i, to stand for an element of I. Thus we are dealing
with a collection of sets A α for α ∈ I. This leads to the following definition.
Definition 1.8
If we have a set A α for every α in some index set I, then
⋃
A α = { x : x ∈ A α for at least one set A α with α ∈ I }
α∈I
⋂
A α = { x : x ∈ A α for every set A α with α ∈ I } .
α∈I