Equivalence Relations and Partitioning in Sets

[Aequiveri]

I have two questions:

i) Does a distinct equivalence relation on a set produce only one possible partition of that set?

ii) Can multiple (distinct) equivalence relations on a set produce the same partition of that set? In other words, given a set S and two distinct equivalence relations ~ and *, is it possible for ~ on S to give the same partition as * on S?

Thanks in advance.

Ae

 

[adriank]

i) Do you mean anything in particular when you say a distinct equivalence relation? What do you mean by produce? Ordinarily an equivalence relation on a set corresponds to the partition on that set consisting of the equivalence classes.

Maybe this will help clarify both i) and ii): There is a one-to-one correspondence between equivalence relations on a set S and partitions of S, which identifies an equivalence relation with its set of equivalence classes. Sketch of proof: If ~ is an equivalence relation on S, let P_~ be the set of ~-equivalence classes of S; show that this is a partition of S. If P is a partition of S, let ~_P be the relation on S such that x ~_P y if and only if x and y are in the same element of P; prove that ~_P is an equivalence relation. Show that these two operations are inverses of each other; that is, P = P_~ if and only if ~ = ~_P.

 

[JonF]

Every equivalence relation corresponds to one partition and every partition corresponds to one equivalence relation. Did they prove the correspondence between equivalence relations and partitions in your class? If so you should be able to spot this in the proof.

 

[arkajad]

Perhaps you should also know that if S is a set and ~ an equivalence relation in S, then the set of equivalence classes is often denoted by S/~.

Read: quotient of S by ~

One day you will certainly meet it.

 

[Aequiveri]

Thank you both for your responses. I now understand.