Why define equivalence relations, posets etc.

[Avichal]

I am studying set theory and I came across various definitions like equivalence relations, partial order relations, antisymmetric and many more. I am aware mathematicians don't care about real life applications but still - why are we defining so many relations? What is the use of defining equivalence relations, posets, maximal element etc.

 

[Bacle2]

All of these are part of the study of general order relations on sets; there are other terms, like maximal elements, supremum, etc., that are used to prove certain results,like the Hahn-Banach theorem,etc. . I don't know if this

is the type of answer you were looking for?

And, BTW, there are applied mathematicians out there interested in applications .

 

[HallsofIvy]

All of mathematics consists in saying that "these things" are the same- in some sense. An equivalence relation. And then we treat all of the thing that are the "same" in that sense as "the same thing"- i.e., an equivalence class. As for "relations" in general, in a very real sense, relations are all there are in mathematics!

 

[Avichal]

The concept of relations and sets is really useful but I don't really understand the use of equivalence relation and other defined things.
Maybe an application of equivalent relation might help

 

[Stephen Tashi]

The concept of relations and sets is really useful but I don't really understand the use of equivalence relation and other defined things.
Maybe an application of equivalent relation might help

Just consider that the symbol "=" and the words such as "equal", "equivalent', "the same" are commonly misused by people who attempt to talk about mathematics without having studied equivalence relations. For example, to say two numbers are "equal" doesn't mean the same thing as saying two sets are "equal". If you say "3 + 8" is "the same" as "11", that's true if you're talking about numbers, but not if you're talking about strings of characters. You can say {3,5,2} is equal to {2,5,3} as a set, but not as a permutation.

If you do computer programming perhaps you have encountered "overloading" of functions or operators. In mathematics the "=" relation is 'overloaded". It is used on many different types of things and has different meanings for each type. However, it has certain universal properties. Hence you should understand what those properties are. (In programming, you need to understand an "abstract class" before you can understand a particular implementation of it.)

There are amusing examples of equivalence relations whose definition is so complicated that the symbol "=" and the word "equal" are not used. For example, the relation of groups being isomorphic is an equivalence relation but we use the word "isomorphic" instead of "equal".

There are examples of two important equivalence relations being defined on the same pairs of objects, which creates the need to use different symbols for the relations. For example the congruence of two numbers mod 7 is an equivalence relation, but it not the same as the ordinary equality relation between numbers. So we must use special notation for congruence mod 7.

 

[Bacle2]

An application of equivalence relation is to general quotients:

quotient groups, vector spaces, quotient topological spaces.

Here we declare elements in the quotient structure A/B to be equivalent

if they satisfy a specific (equivalence) relation . Like Halls said, to the

effect of what we are doing , we only care about certain specific aspects

of the elements, and nothing else.

Also, when we talk about, say, the set "multiples of n" and its

properties, we are talking of the set {3n: n is natural}. All of these

elements ...,3,6,... are equivalent under a~b iff. 3|(a-b).

 

[Hurkyl]

The concept of relations and sets is really useful but I don't really understand the use of equivalence relation and other defined things.
Maybe an application of equivalent relation might help

You have the concept of a rational number. But 'what' are they? Or a better question: how can I represent them in terms of things I already know?

Well, that's easy. We learned in elementary school a number of ways to write rational numbers. A common way is that they are a pair of numbers: a numerator and a denominator, which are both integers!

Oh, but wait: 2/4 and 3/6 are the same rational number. So just representing a pair of numbers is not good enough: we must have an additional notion of when two different pairs of numbers really represent the same rational number. That is, when they are are equivalent: i.e. we need an equivalence relation.

In this case, the relevant equivalence relation is that a/b ~ c/d if and only if ad = bc.



It is very common that one is interested in some sort of object, and one knows how to represent that object as a different sort of object, but two different representations are really the same object. So one needs to also find an equivalence relation on the representations, in order to use this representation to help understand the thing you're really interested in!


The converse happens too: we understand two different sorts of objects, and we discover that we can use one sort of object to represent the other sort. Sometimes, the relationship between the sorts objects is best understood by figuring out the equivalence relation the representation defines.

This last paragraph may be a little muddled, so let me give an example. It's going to the the same example as before, but from a different angle.

We understand rational numbers. And we also understand pairs of integers with the second number nonzero. And we know there is a relationship between them: the pair (x,y) yields the rational number x/y.

But how do we do computations with rational numbers? Most of the time, it involves representing x/y by a corresponding pair (x,y), and doing computation with the pairs. So for the sake of computation, it is very helpful to know that the pairs (x,y) and (s,t) correspond to the same rational number if and only if xt = ys.