Investiga tion 5
Equivalence Relations and
By the end of this investigation, you should be able to give precise and thorough
answers to the questions listed below. You may want to keep these questions in mind
to focus your tho ughts as you complete the investigation.
• What is a congruence class, and what are some properties of congruence classes?
• What is an eq uivalence re lation, and what ar e some strategies for proving the prop-
erties that characterize equivalence relation s?
• What are equivalence classes, and how do the equivalence classes corresponding
to an equivalence relation divide the underlying set in to sub sets?
• What is Z
, and wha t arithm etic axioms hold within Z
• What is a binary operation, and what does it mean for a binary operation to be
• What ar e zero divisors and units in Z
? How are zero divisors an d units related to
solving line a r equations?
Preview Activity 5.1. Wh en working with large sets of objects, it is often useful to group these
objects accor ding to some common a ttribute or property. For instance, in a cooler containing 100
cans of soft dr inks, there may be 30 can s of Coke, 30 cans of Pepsi, and 40 cans of 7 Up. If
someone wanted to drink a can of Coke, they probably would not care exactly which can of Coke
they pu lled out of the cooler. In other words, they would probably consider all of the different cans
of Coke to be indistinguishable, or equivalent. This same kind of grouping can be applied to a
set of mathematical objects by deﬁning an equiv alence relation. In this p review activity, we will
investigate how congruence can be used to deﬁne such a relation on the integers.
(a) For every integer a, let [a]
denote the set of all integers that are congruent to a modulo 3.
Using the roster method,
list the elements in 
(b) Repeat part (a) for 
, and 
. Do you notice anything about the relation-
ships between these sets?
(c) What is the remainder when 73 4 is divided by 3? Which, if any, of the sets 
To specify a set using the roster method, we simply list the elements of the s et between braces, as in Deﬁnition 1.2 on