# Equivalence Relation and a Function

1. Apr 28, 2009

### philbein

1. The problem statement, all variables and given/known data

Suppose that A is a nonempty set and R is an equivalence relation on A. PROVE that there is a function f with A as its domain such that for x and y in A, xRy (x is related to y) if and only if f(x)=f(y)

2. Relevant equations

Equivalence relations are relations that are reflexive, symmetric, and transitive.

Theorem: If R is an equivalence relation on a set A. Then, the equivalence classes of R form a partition of A. (The converse is also true).

3. The attempt at a solution

My guess on this is that we are supposed to use the theorem with relations, and partitions that I stated above. Where I get confused is where do these functions tie in, and I am completely clueless on where to get started here. Any ideas would be great. Thanks.

2. Apr 28, 2009

### Billy Bob

Let E be the set of equivalence classes of R on A.

Define f from A to E by letting f(a)= (what do you think? there is really only one natural choice).

Show that for all x and y in A, xRy if and only if f(x)=f(y).