Computability and Logic

12.2. EQUIVALENCE RELATIONS 143
Namely, for all a, b, c in X we must have the following:
(E1) Reflexivity: aEa.
(E2) Symmetry: IfaEbthen bEa.
(E3) Transitivity: IfaEband bEcthen aEc.
A relation with these properties is called an equivalence relation on X.
One way to get an equivalence relation on X is to start with what is called a
partition of X. This is a set of nonempty subsets of X such that the following hold:
(P1) Disjointness:IfA and B are in , then either A = B or A and B have no
elements in common.
(P2) Exhaustiveness: Every a in X belongs to some A in .
The sets in are called the pieces of the partition.
Given a partition, define aEb to hold if a and b are in the same piece of the
partition, that is, if, for some A in , a and b are both in A. Now by (P2), a is in
some A in . To say a and a are ‘both’ in A is simply to say a is in A twice, and
since it was true the first time, it will be true the second time also, showing that
aEa, and that (E1) holds. If aEb, then a and b are both in some A in , and to say
that b and a are both in A is to say the same thing in a different order, and is equally
true, showing that bEa, and that (E2) holds. Finally, if aEband bEc, then a and
b are both in some A in and b and c are both in some B in . But by (P1), since
A and B have the common element b, they are in fact the same, so a and c are both
in A = B, and aEc, showing that (E3) holds. So E is an equivalence relation, called
the equivalence relation induced by the partition.
Actually, this is in a sense the only way to get an equivalence relation: every
equivalence relation is induced by a partition. For suppose E is any such relation;
for any a in X let [a] be the equivalence class of a, the set of all b in X such that
aEb; and let be the set of all these equivalence classes. We claim is a partition.
Certainly any element a of X is in some A in , namely, a is in [a], by (E1). So (P2)
holds. As for (P1), if [a] and [b] have a common element c,wehaveaEcand bEc,
and having bEc, by (E2) we have also cEb, and then, having aEcand cEb,by
(E3) we have also aEb, and by (E2) again we have also bEa. But then if d is any
element of [a], having aEdand bEa, by (E3) again we have bEd, and d is in [b]. In
exactly the same way, any element of [b]isin[a], and [a] = [b]. So is a partition,
as claimed. We also claim the original E is just the equivalence relation induced by
this partition . For along the way we have shown that if aEbthen a and b belong
to the same piece [a] = [b] of the partition, while of course if b belongs to the same
piece [a] of the partition that a does, then we have aEb,soE is the equivalence
relation induced by this partition.
We can draw a picture of a denumerable model of Eq, by drawing dots to represent
elements of X with boxes around those that are in the same equivalence class. We
can also describe such a model by describing its signature, the infinite sequence of
numbers whose 0th entry is the number (which may be 0, 1, 2, ..., or infinite) of
equivalence classes having infinitely many elements and whose nth entry for n > 0is