Why Read This Book? - Index of

72 Chapter 3 Sets and Their Properties
In Definitions 3.3.3 and 3.3.4, we will define precisely what we mean by this sort
of union and intersection.
We can go even further. It is conceivable we might need to work with infinitely
many sets {A1,A2,A3,...} that we might want to index with the positive integers.
For example, if we use the familiar interval notation
[a, b] ={x : x ∈ R and a ≤ x ≤ b}
we might talk about the family of intervals F ={An}n∈N, where An =[0, 1/n].To
form the union or intersection of a family of sets indexed by the positive integers,
we could use notation like that in Eqs. (3.11) and (3.12).
∞�
An and
n=1
∞�
n=1
or we could write something like
�
An and �
n∈N
n∈N
An
An
(3.13)
(3.14)
where by (3.14) we understand that n is allowed to take on all values of the indexing
set of natural numbers.
The notation in (3.14) is handy when the indexing set is more complicated than
the positive integers and does not allow us to think of some index variable n starting
at 1 and progressing sequentially off to infinity. For it is conceivable that any set A
can index a family of sets. We can then address individual sets in the family as Aα,
where α ∈ A, and denote the family F ={Aα}α∈A. Union and intersection could
then be written as
�
Aα and �
α∈A
Example 3.3.1 One important contribution of the German mathematician
Richard Dedekind (1831–1916) is a rigorous foundation of the set of real numbers.
He employed what is now called a Dedekind cut, whereby the set of rational numbers
is “cut” into two pieces. Specifically, for a real number r, Dedekind worked
with sets of the form
α∈A
Aα
Ar ={x : x ∈ Q and xr} (3.16)
For example, although √ 2 is irrational (as you will see in Section 3.10), it certainly
makes sense to talk about A √ 2 , the set of all rational numbers less than √ 2. The
real numbers form the index set for the families {Ar}r∈R and {Br}r∈R. �