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Equivalence Relations 213
20. A bus company is trying to decide what bus routes to run among the cities
∗
in the set C ={San Francisco, Chicago, Dallas, New York, Washington
D.C.}. Their routes will be represented by a relation B on C,asinthe
example in the text. The company wants to make sure you can get from
any city in C to any other city in C, so they want to make sure that the
transitive closure of B is C × C. Let F ={B ⊆ C × C | the transitive
closure of B is C × C}. However, they don’t want to run any bus routes
unnecessarily, so they want the relation B to be a minimal element of F.
(As usual, we mean minimal with respect to the subset ordering on F.
You will have to work out what this means, according to the definition of
minimal in Section 4.4.)
(a) Find a minimal element of F.
(b) Does F have a smallest element?

4.6. Equivalence Relations

We saw in Example 4.3.3 that the identity relation i A on any set A is always
reflexive, symmetric, and transitive. Relations with this combination of prop-
erties come up often in mathematics, and they have some important properties
that we will investigate in this section. These relations are called equivalence
relations.

Definition 4.6.1. Suppose R is a relation on a set A. Then R is called an
equivalence relation on A (or just an equivalence relation if A is clear from
context) if it is reflexive, symmetric, and transitive.

As we observed earlier, the identity relation i A on a set A is an equivalence
relation. For another example, let T be the set of all triangles, and let C be the
relation of congruence of triangles. In other words, C ={(s, t) ∈ T × T | the
triangle s is congruent to the triangle t}. (Recall that a triangle is congruent
to another if it can be moved without distorting it so that it coincides with the
other.) Clearly every triangle is congruent to itself, so C is reflexive. Also, if
triangle s is congruent to triangle t, then t is congruent to s,so C is symmetric;
and if r is congruent to s and s is congruent to t, then r is congruent to t,so C
is transitive. Thus, C is an equivalence relation on T.
As another example, let P be the set of all people, and let B ={(p, q) ∈
P × P | the person p has the same birthday as the person q}. (By “same
birthday” we mean same month and day, but not necessarily the same year.)
Everyone has the same birthday as himself, so B is reflexive. If p has the same
birthday as q, then q has the same birthday as p,so B is symmetric. And if p has