Relations, Part Ia
Equivalence elations
and Partial Orderings
CHAPTER 7
In the next two chapters we study three related, yet diverse, mathematical
concepts. Equivalence relations and partial orderings are studied in the
present chapter, while the topic functions/mappings is covered in Chapter
- You have undoubtedly had considerable experience with functions, but
the terms "equivalence relation" and "partial ordering" are likely to be un-
familiar. In spite of this, most of you will probably feel more "at home"
with the latter two concepts than you might anticipate, while feeling less
familiar than expected with the approach to functions and mappings in
Chapter 8. An abstract treatment of functions/mappings, although dealing
with a familiar concept, has a strikingly different emphasis from that seen
in precalculus and elementary calculus courses. On the other hand, equiva-
lence relations and partial orderings, even though probably new to you as
concepts, generalize familiar mathematical relationships.
The most basic example of an equivalence relation is the relationship
"equals." The relationships of equality between numbers, equality between
sets, and indeed equality between any kinds of objects, are all examples of
equivalence relations. More generally, equivalence relations are the mathe-
matician's way of describing situations in which two objects can, in some
sense, be considered and treated as "the same." Viewing different objects
as indistinguishable, from some specific vantage point, is common, both
within and outside mathematics. As one example, high school geometry
students with no knowledge of equivalence relations find it natural to regard
two congruent triangles as identical in the context of Euclidean plane ge-
ometry. At even more elementary levels, students are trained to regard pairs