Equivalence Relation and a Function

  • Thread starter philbein
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Homework Statement



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)

Homework 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).


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.
 

Answers and Replies

  • #2
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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).
 

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