Equivalence Relation and a Function

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SUMMARY

The discussion centers on proving the existence of a function f from a nonempty set A to its equivalence classes E, such that for any elements x and y in A, x is related to y (xRy) if and only if f(x) equals f(y). The key concepts include equivalence relations, which are defined as reflexive, symmetric, and transitive, and the theorem stating that equivalence classes form a partition of A. The solution involves defining f in a manner that aligns with these properties, specifically by mapping elements of A to their respective equivalence classes.

PREREQUISITES
  • Understanding of equivalence relations and their properties (reflexivity, symmetry, transitivity).
  • Knowledge of set theory and partitions.
  • Familiarity with functions and their definitions in mathematical contexts.
  • Basic proof techniques in mathematics, particularly in relation to functions and relations.
NEXT STEPS
  • Study the properties of equivalence relations in detail.
  • Research the concept of partitions in set theory.
  • Learn about functions and their mappings in mathematical proofs.
  • Explore examples of equivalence classes and their applications in various mathematical contexts.
USEFUL FOR

Students of mathematics, particularly those studying abstract algebra or discrete mathematics, as well as educators looking to enhance their understanding of equivalence relations and functions.

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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.
 
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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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