8.3 CARDINAL NUMBER OF A SET 277(d) Suppose a E A - im g or b E B - im f. What is the appropriate conclusion
about ancestry in both these cases?
(e) Suppose an element a E A has an odd number of ancestors. What must be true
of the element f (a) of B? Formulate three additional analogous statements.
(f) Use the preceding mappings f and g to create a bijection between A and B.
8.3 Cardinal Number of a Set
This article might be considered an extension of Chapter 1, especially Article
1.5, because we return to the topic of sets and, in particular, to questions
related to the of a set. As you will soon see, however, the flavor of
the material in this article differs greatly from that in Chapter 1 for two
principal reasons: (1) Unlike the earlier treatment, we set down in this article
rigorous definitions of finite set and infinite set. (2) Then we focus on the
infinite set category. You may recall that, in Article 1.5, we developed for-
mulas for counting the elements of finite sets. Implicit in this study was
the understanding that two finite sets can, in many cases, be distinguished
from each other (i.e., can be established to be different sets) purely on the
basis of their differing number of elements. Any thought you may have
given at that stage to infinite sets was probably with an equally certain,
although again only implicit, assumption that there is no way of differen-
tiating the sizes of infinite sets. This assumption is incorrect! Indeed, the
single most important idea of this article is that relative sizes of inJinite sets
can be distinguished in a mathematically satisfying and useful way. As one
example, we will see that the familiar sets N, Z, and Q can be distinguished
from R and C with respect to "size," but N and Q, for instance, cannot be
so differentiated! The existence of different levels of infinity rests on the
theory of infinite cardinal numbers, the creation of which (by Georg Cantor)
inspired the development of much of modern set theory and formal logic.
The theory of infinite cardinal numbers is not only a fundamental tool
in modern mathematics at the graduate and research levels, but since its
creation in the late nineteenth century, has been widely celebrated as
both an example of the creative genius of the human mind and a prime
example of the aesthetic appeal of mathematics at its finest! Much of the
theory is beyond both the level and intent of this text; hence our objective
is limited to giving you some idea about the theory, and providing in the
process a foundation for possible further study.
NUMERICAL EQUIVALENCE OF SETSWe begin our consideration of cardinality of sets by restating formally a
definition alluded to in Article 8.2.
DEFINITION 1
Let A and B be sets. We say that A and B are numerically equivalent, denoted
A r 8, if and only if there exists a one-to-one mapping f: A -+ B of A onto B.