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We have
shown a close relationship between the concept of one-to-one
correspondence and the idea of the number of elements in a set, called
the cardinality of a set. (See the counting of the insects above.) This
exploration has led us to the following definitions relating the sets of
natural and whole numbers to finite sets. Further, we note that this
relationship is closely related to how small children learn to count.
Definitions.
The set of
natural numbers (or
counting numbers) is the set
{1, 2, 3, …}.
The set of whole numbers
is the set {0, 1, 2, 3, …}.
Definition.
Let
A be a finite set. If set
A is equivalent to the subset
{1, 2, 3, …, N} of the
natural numbers, we say that the
cardinal number for set A
is N.
Notation:
n(A)
= N. The cardinal number for
the empty set, Ø,
is 0, i.e.,
n(Ø
) = 0.
Let C = {#, $, %, &}.
Show n(C)
= 4.
|
#
$
%
& |
|
Hence,
C
~ {1, 2, 3, 4} and
n(C)
= 4 since a 1-1 correspondence can be
setup between
C
and {1, 2, 3, 4}. |
You probably learned
the cardinal number zero, 0, much later in life, well after you learned
how to count. This is also true in the history of humans. The cardinal
number zero was invented much later than the natural numbers.
Additional Notes.
We
will not give precise definitions for the terms
finite and
infinite. We will consider a
finite set to be a set that
has a cardinal number that is a natural number and an
infinite set as a set that is
not finite. Think of a finite set as a set that has a limited number of
elements and an infinite set as a set that has an unlimited number of
elements.
Side Note.
The cardinal number for any set equivalent to the set of all the natural
numbers is אo
read as
aleph-nought. Aleph is a letter in the Hebrew alphabet.