Re: equivalence classes
- From: fernando revilla <frej0002@xxxxxxxxxxxxxxxxxx>
- Date: Mon, 26 Sep 2005 10:13:23 EDT
> Hey,
>
> I'm looking for some help with something that is
> probably very simple,
> but I just can't seem to figure it out.
>
> I've got the basic idea down of proving that a
> relation is an
> equivalence relation, but I'm having a hard time with
> describing
> equivalence classes. For example, "Given x, y are
> elements of [0,
> infinity), x ~ y means y = sq(x^2).
>
> Now, it seems to me that for each equivalence class
> of, say, x ~ 0 there
> is only one x that is an element of [0, infinity).
> Thus it would seem
> that there are an infinite number of equivalance
> classes. However, my
> book seems to make it out that there are always
> finite numbers of
> equivalence classes for any set, X, such that they
> can be described,
> say, for my homework assignment. Am I missing
> something or are there
> cases where there are infinite equivalence classes?
> If there can be
> infinite equivalence classes, how does one usually
> site this?
>
> Thanks.
>
> AEM
Although an equivalence relation in a non empty class
A may be defined by comprehension, this is equivalent
to stating that in the class A we have a partition i.e.
there exists a class C of non empty subclasses A_k
of A satisfyng:
1.- Every pair of elements of C are disjoint.
2.- C is exhaustive, i.e. the union of all the elements
A_k is A.
Perhaps in this way it is easier to see all the posibilities,
every A_k is an equivalence class. However the most
important thing about equivalence relations is the fact
that allow to define new concepts, for instance the
abstract concept of red, blue,yellow,..
Fernando.
.
- References:
- equivalence classes
- From: Anon E . Mouse
- equivalence classes
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