Talk:Equivalence class

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I think some mention should be made of representative elements of an equivalence class and reduction relationships that lead to such representations.

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My math is old, rusty and unused. Therefore I find the article a little confusing in some places. In particular the first paragraph of Properties: "that any two equivalence classes are either equal or disjoint."

I think it would be great if someone would add correct, simple examples. For example by elaborating the colored cars example.

As far as I can tell from the equivalence_relation page, there can be different equivalence relations leading to different equivalence classes, which may overlap between two equivalence relations. So the quotation above perhaps should continue "under the same equivalence relation".

Even the example given can be viewed in two ways: either partitioning the set of cars into a partition per color (x~y == color(x) = color(y)) , or one partition of green cars and one of non-green cars (x~y == color(x) = green iff color(y) = green). Is it somehow interesting that the equivalence class "red cars" is also an equivalence class within the subset of non-green cars?

It would also seem that the number of possible different equivalence classes over all possible equivalence relations on a non-empty set A is 2^|A|-1. But there could be fewer. Consider this set of cars (color,brand,model): { "blue Ford Galaxy", "blue Ford Focus", "silver Opel Zafira", "red Opel Astra", "red VW Polo" }. Partitioning on color and partitioning on brand both yield the set of blue Fords as one of the equivalence classes, so there are less than the 31 possible distinct equivalence classes, is that correct?

--Lasse Hillerøe Petersen 09:35, 17 July 2007 (UTC)

Who is the Wikipedia's audience?

There are (IMO) very few topics in the Wikipedia more important to more subjects than 'equivalence class'.

This articles raises the question of who articles are written for. Geology articles, I can assure you, are simplified to the level of inaccuracy. It was to this article I was about to refer discussions on classifying the sciences. My claim was that a physical science like chemistry was more abstract than a natural science like geology, because the former studied philosophically identical objects, like the element gold, whereas geology studied equivalence classes of specimens, like the mineral gold (which varies in properties). Were I to relate objective properties, equivalence, equivalence classes, abstract & concrete, a general reader would come here for a clear explanation of 'equivalence class'.

Classification founds science and many other unrelated fields. Perhaps this article could be broadened in scope, lengthened, offer the reader motivations by examples (as the previous discussion has done), proceed from the concrete to the abstract, and attempt to use intuitive English rather than offer an explanation in purely formal, symbolic logic. There would be plenty of room for that in a mathematics section. A reference would be nice. Geologist (talk) 09:01, 20 December 2007 (UTC)

Order?

[...] then the order of X/~ is the quotient of the order of X by the order of an equivalence class.

• Why not say cardinality instead of order?
• Can it be that X/~ is always a partition of the set X?

Thanks, --Abdull (talk) 10:44, 4 July 2008 (UTC)

So let me explain!

I recently made an edit concerning canonical representatives which was immediately (and absolutely correctly) reverted by David Eppstein. Let me explain what happened. I had recently added an example (rectangles) of equivalence classes that were not arithmetic in nature, but more mathematical than cars. I was then thinking that a second example (again, a non-arithmetic one) of canonical representatives would be useful, so I turned again to the rectangles of equal area example. Here's what went wrong. When I put the example in, I added the innocent phrase, "in a plane", thinking that that would make the example more concrete ... but what it really did was to change the set I was dealing with. The set I had in mind was that of "abstract" rectangles, whose only attributes are width and length, and not the set of rectangles embedded in a plane with their additional attributes of position and orientation. This doesn't make much difference as far as the example is concerned (either set gives a good example), but only the former has canonical representatives. At issue is what to do now. I could go in and fix the description to get the example of canonical representatives that I was aiming for, but I am wondering whether it is worth the effort to do so. Opinions? Bill Cherowitzo (talk) 22:32, 17 March 2014 (UTC)

Why you do not choose the example of the circles, that are well defined by their radius and the position of their center? Under congruence (or equivalently displacements), a canonical representative is the circle of the same radius centered at the origin. This example has both advantages of introducing the important class of the equivalences classes under geometrical transformations, and to explain the jargon of "the unit circle", which is the canonical representative of the class of circles of radius one.D.Lazard (talk) 08:30, 18 March 2014 (UTC)

Are you assuming that the origin itself has been chosen canonically somehow? Because with a coordinate system fixed in place, such as Cartesian coordinates, squares are also easy to make canonical. But my understanding is that most axiomatic treatments of the Euclidean plane do not distinguish any particular point as the origin. —David Eppstein (talk) 16:06, 18 March 2014 (UTC)

I agree, I should have written "If the origin of a Cartesian coordinate system has been chosen, under congruence (or equivalently displacements), a canonical representative is the circle of the same radius centered at the origin". Almost the same is true for the square. However, one needs not only the choice the origin, but also of the direction of the axes. Also there are two choices for the canonical representative: the square centered at the origin or the square in the first quadrant, with a corner at the origin.

About your second sentence: I disagree with "most axiomatic treatments": All the definition of the Euclidean plane, either axiomatic or through linear algebra have been shown to be equivalent, and do not depend on the choice of the origin. The common identification of a Euclidean plane with R² is, in fact an abuse of language: It is the pair of a Euclidean plane and an orthonormal Cartesian coordinate system, which may be identified with R². D.Lazard (talk) 16:55, 18 March 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Equivalence class/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

A very important subject. Much work needed. And references too. Geometry guy 00:11, 17 June 2007 (UTC) Isn't the equivalence class fundamental to classification; and doesn't classification found many if not every field of study: not just those listed above? Geologist (talk) 07:40, 20 December 2007 (UTC)

Last edited at 07:40, 20 December 2007 (UTC). Substituted at 02:03, 5 May 2016 (UTC)

Lets relocate one section to the stub article quotient

The Wikipedia article Quotient is (like this article) in rather poor shape. Wiki project mathematics rates the article Quotient as "High" importance and "Stub" quality. That article has a lot of potential as a unifying overview of quotient structures of every kind (quotient topology, quotient group, quotient vector space).

I propose that section "Quotient space in topology" has much more detail on general quotient structures than this article warrants, and that the content of that entire section should be migrated to the article quotient. How do you guys feel?

Norbornene (talk) 02:37, 18 August 2016 (UTC)

Identification map

"Identification map" redirects to this article and yet, there is no mention of "identification map" in the article. Am I supposed to think that "identification map" is the same as "equivalence class"?

62.44.100.59 (talk) 17:02, 13 March 2017 (UTC)

The redirect was wrong. I have fixed it. Nevertheless, this is a very unusual terminology (only two WP articles link to this redirect). D.Lazard (talk) 18:25, 13 March 2017 (UTC)