228 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7of fractions such as $ and & as the same for purposes of calculation. In
a nonmathematical sphere everyone is familiar with the idea that two coins
minted for circulation are the same if and only if they are of the same denom-
ination, for example, both pennies, both quarters, and so on, and "differ-
ent" otherwise. Hence, although two dimes are different as physical entities
(and may, e.g., be different when viewed through the eyes of a coin collector),
they are the same with respect to their value as money, the criterion we
normally have in mind when dealing with coins. This fact, in turn, affects
our attitude toward coins; when dealing with coins purely as money, we
care not about individual coins, but only about classes of coins (e.g., the
class of nickels, the class of half-dollars, etc.). The primary role of an indi-
vidual coin is that of arbitrary representative of the class containing it.
The mathematically rigorous reason that we can "identify" distinct ob-
jects in examples such as the preceding three is that there is an equivalence
relation implicitly underlying each. Because we are able to regard distinct
objects as indistinguishable by means of equivalence relations, it is possible
to study sets of mathematical objects by dealing with subsets consisting of
elements identified with one another by the relation, known as equivalence
classes, rather than with individual elements. Sets of equivalence classes,
in turn, are fundamental to some of the most important constructions in
mathematics. In Chapter 10 we deal with the questions, "What are the
rational numbers?" and "What are the real numbers?' using an approach
based on equivalence classes.
The concept of partial ordering is a generalization of the relationship
"less than or equal to" on the set of real numbers. Whenever a partial order-
ing on a set of objects has been defined, some idea of "relative size" of some
or all of the objects in the set is implied. The notion of partial ordering is
the foundation of a number of theories falling under the heading "ordered
algebraic structures" in advanced mathematics. Our treatment gives a very
brief introduction to such theories.
The common thread linking the concepts of equivalence relation, partial
ordering, and function/mapping is the notion of relation between two sets.
We begin by focusing on that concept.
Relations
The concept of relation from a set A to a set B is based on the concept of
ordered pair (x, y) and, more specifically, the idea of the cartesian product
A x B of two sets A and B. You may wish to reread some of the relevant
material in Article 1.2 and recall Exercise 1, Article 5.2; Exercise 4, Article
6.1; and Exercise 1, Article 6.2. Leaving aside, as in Chapter 1, a formal
definition of "ordered pair" (given in Exercise 10, Article 4.1), we recall here
the criteria for equality of ordered pairs and for membership of an object
in the cartesian product of two sets.