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