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What is the purpose of the relationships of equivalence?

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What is the purpose of the relationships of equivalence?

Postby elkan » Tue Dec 25, 2012 8:10 pm

In logic, what is the purpose of the relationships of equivalence?
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What is the purpose of the relationships of equivalence?

Postby edmund » Tue Dec 25, 2012 8:18 pm

You should be more clear on what you mean by "purpose" in this context. I think it's probably a mistake to say that equivalence relations themselves have any purpose. I tend to doubt that the assertion that they do makes any sense, but I feel confident that if they somehow did have purposes, those purposes would be closed to our knowledge from this side of eternity.

Although there are many purposes that people can have in wanting to find or point out equivalences, these purposes are too diverse to describe or to helpfully be thought of as one purpose. Having said this, all of their well-motivated purposes can, at least at a very high level of abstraction, be thought of as having one general purpose - more on that later.

When there's an equivalence relation between x and y, any or all of the following are true:

- x and y are exactly similar in some regards

- x and y share some property, z, in common (whatever other properties they may or may not have)

- x and y are members of a common subclass (or part) of some, possibly other, class (or part)

In fact the above all amount to pretty much the same thing.

A closely related assertion is that:

- x and y share a common subclass (or part)

One basic purpose - "the" purpose, as mentioned above - that logicians, or anyone, might have in finding or pointing out an equivalence relation is to divide up a class they're interested in into mutually exclusive, jointly exhaustive subclasses or to show that this can be done. This larger class can then be investigated with regards to how it is divided up and information, sometimes surprising, can be learned about it. I'll give examples shortly.

When we talk about mutually exclusive and jointly exhaustive of a class, "meje", we mean to say that there's some process which assigns every member of that class to some subclass (jointly exhaustive) and that no member is assigned to more than one subclass (mutually exclusive).

Imagine collecting a child's toys into dolls and cars and blocks and puzzles and games and however many or few groups you want. Perhaps you want to divide them up into just dolls and nondolls (two subclasses) or even to just count them, "dividing them" all up into toys (one class). Each of these divisions, however many or few, divides the class of toys into meje subclasses of "same kind of thing" or "same kind of toy". However you go about it, you'd learn something about the numbers and/or kinds of toys the child has and could use that information to get still more information. Perhaps you want to know the ratio of stereotypically male toys to female, or electronic toys to nonelectronic; it could be anything.

Another much more sophisticated example goes on every day with online advertisers who use information acquired from tens of millions of user clicks to divide those users up into likely groups for targeted advertising, political e-mails, etc. The same basic principles are used in dna analysis to identify which gene segments are most likely to have certain functions. The techniques have even been used in law enforcement and anti-terrorism measures, automated stock market trading, language analysis software, and more. Basically computers just crunch huge sets of data looking for convoluted patterns of similarity - "equivalences". If they find enough distinct respects of similarity they've discovered something. "All who do this, this, and this, also do this." is how the reasoning might go.

So you can see why I wrote that, although at a very high level of abstraction people looking for or at equivalence classes all have the same purpose, that level of abstraction is so high as to not be particularly meaningful. Parents cleaning a room, online marketers, politicians, geneticists, police, and stock traders are not all doing the same thing, even though, in some sense, they really are.

You might not be able to follow all of what's in the following links but get what you can out of them. The first two deal with what an equivalence relation is. The next two deal with finding complicated equivalence relations and the uses put to such. I should note that what's actually being found in these cases are not true equivalence relations but similarity relations. Since similarity relations need not be transitive like equivalence relations, my discussion of them is, strictly speaking, a bit off point. However there's a strong enough connection between the two notions that it's helpful to discuss one to elucidate the other: similarity classes have blurry edges, equivalence classes sharp.




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