Why Read This Book? - Index of

3.7 Equivalence Classes and Partitions 101
Example 3.7.6 From Example 3.6.2, the equivalence class of Auckland is the
set of all cities with roads leading to Auckland, that is, all cities on the north island
of New Zealand. It is the same as the equivalence class of Wellington. �
Example 3.7.7 For equivalence mod 6 from Exercise 3.6.18, given any integer x,
[x] ={y ∈ Z : y − x = 6k, for some k ∈ Z}
Thus [x] is the set of all integers y of the form y = 6k + x, where k takes on all
possible integer values. �
EXERCISE 3.7.8 From Example 3.7.7, describe [−4] by listing some of its elements.
How many distinct equivalence classes are there in this example? Describe
all of them by listing some elements from each equivalence class.
EXERCISE 3.7.9 From Example 3.6.14, how many assignments are in the
equivalence class of the assignment in Figure 3.10?
In all the preceding examples of equivalence relations, the equivalence classes
seem to be a basis of a partition of the set. This is no coincidence, of course.
Theorem 3.7.10 says that properties E1–E3 can be used to demonstrate that the
family of equivalence classes satisfies P1–P3. Most details are left to you as an
exercise.
Theorem 3.7.10 Suppose ≡ defines an equivalence relation on a set S. Define
F ={[x] :x ∈ S }. That is, F is the collection of all equivalence classes generated
by letting x take on all values in S. Then F is a partition of S.
Proof. We verify that properties P1–P3 from Definition 3.7.1 are satisfied.
(P1) Pick any [x] ∈F. Since x ≡ x, it follows that x ∈[x]. Thus [x] is not empty.
(P2) Pick [x], [y] ∈F and suppose [x]∩[y] is not empty ....
(P3) Since every set in F is defined to be a subset of S, we have ∪x∈S[x] ⊆S by
Exercise 3.3.13. So we must show ⊇. Pick y ∈ S ....
EXERCISE 3.7.11 Complete parts P1–P3 of the proof of Theorem 3.7.10.
Theorem 3.7.10 is very important for the following reason. If some form of
equivalence is defined on a set, and it can be verified that this definition is an
equivalence relation, then Theorem 3.7.10 guarantees that the set is automatically
partitioned into equivalence classes. We can then think of any element of
a particular equivalence class as being representative of all elements in its class.
Furthermore, we can think of this equivalence in precisely the same way we think
of equality, where two equivalent elements are, in many senses, interchangeable.