Elementary Abstract Algebra- Examples and Applications, 2019a

14
Equivalence Relations and
Equivalence Classes
In the previous chapter we introduced the abstract concept of group,
which was defined in terms of properties that we’d seen in many previous
examples. We may say that “group” is a generalization which includes many
Generalizations like this play a key role in mathematics: if we can prove that
a particular mathematical structure is a group, then all of the general group
properties must also be true for that particular structure. In this way, we
learn a great deal about the structure with very little effort.
In this chapter we introduce another generalization: the idea of a mathematical
relation, which generalizes the concept of function as formally defined
in Definition 6.2.11. We explore various types of relations and their
properties, and use these new ideas to envision modular arithmetic from a
different perspective. The new concepts that we introduce in this chapter
are foundational to the notions of coset and conjugacy class, two key grouptheoretic
structures which play central roles in group theory (as we shall see
in subsequent chapters).
This chapter is based on material by D. and J. Morris, which was extensively
revised and expanded by Mark Leech.
14.1 Binary relations
Recall that according to Definition 6.2.11, any function f : A → B can be
represented as a set of ordered pairs. More precisely, each element of f is
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