Something I was thinking about

seriously nobody?

>>9077427
Wrong board, /sci/ is for QI and religion, you have a better chance on /pol/ or /b/ (I'm serious, especially for /pol/) if you have a question that require more than high school knowledge.

To answer partly to your question you need that :
-f(a) R g(a) for every a in A
-f(a) R g(b) and f(b) R g(a) for every a,b in A

>>9077817
thanks man, guess I should have known better haha
naturally you also need that if f(a) R g(b) and f(b) R g(c) then f(a) R g(c) for every a, b and c in A.

But I'm actually looking for less trivial requirements for f and g. It's a tough question I guess, but I can't think of more requirements or properties f and g might have.

>>9077832
>naturally you also need that if f(a) R g(b) and f(b) R g(c) then f(a) R g(c) for every a, b and c in A.
Actually you don't need this part, indeed if you have aSb and bSc then f(a) R g(b) and f(b) R g(c)
but since f(b) R g(b) and f(c) R g(c)
you have f(a) R g(c) =>aSc

So the 2 properties I listed are sifficient.

>>9077845
hey that's pretty clever!

>>9077817
This is right.

>>9077427
Think about this in terms of equivalence classes and it's easy to see that f and g take elements from one equivalence class and map them to another (say from equivalence class B to class C) keeping in mind that the domain and codomain for f and g are the same. That's all you really need and then the relations hold.
>>9077817
Seriously? /pol/ threads are usually pretty shitty too.

>>9078121
could you please elaborate? maybe give some examples to cases that work and cases that don't?

>>9078315
Consider the positive integers and the relation a = b mod 3 (a is related to b if the difference is divisible by 3) this clearly forms an equivalence relation and partitions in the positive integers into X={0,3,6...}, Y={1,4,7...}, and Z={2,5,8...}. Now consider some maps f and g such that f and g both take elements from X to Y, Y to Z, and Z to X (how they do this is irrelevant). Clearly f(a) and g(b) are related by the established equivalence relation if and only if a and b are in the same equivalence class, but we defined our functions to do this, so we are done.

>>9078362
Thanks a lot!
Now the real question is - what types of transformations display these sorts of behaviors (i.e always map a and b to the same classes).
I imagine this question is even harder to answer, and depends on the nature of the relation and the group A.

>>9078389
The way I'd think about it is this: if you look don't look at the set and the equivalence relation but instead looked at the equivalence classes themselves then that means that from this perspective the maps f and g are identical (they act on equivalence classes in the same fashion) this provides a general way of looking at it and reveals that f and g may mix up elements within an equivalence class but never swap elements from different equivalence classes.

File: MR-360289-778461-16.jpg (522KB, 1523x2163px) Image search: [Google]

522KB, 1523x2163px

>>9077817
>you have a better chance on /pol/
You realize the IQ and religion threads are posted by /pol/tards?

>reformulating OP's question and calling it a partial answer
you need to go back

>>9078469
Teah, that seems pretty reasonable. I was thinking more about possible algebraic properties of f and g.
The end goal is to charactarize a certain class of transformations that preserve equivalency and understand their properties - i.e, to find a more general way to tell if two transformations preserve equivalency, without having to look specifically at each and every transformation, just like you can tell if a transformation is linear without going through the full process every time.

>>9079545
I imagine that f and g gain the algebraic properties of the underlying structure, in the example I provided (integers mod 3) there exists a group structure that I believe can be used to show that the functions also inherent a group structure since multiplication within the equivalence class maps the element back into that very same equivalence class (3*6=18 = mod 3). So the algebraic properties of the functions are those that get "lifted" from the underlying algebraic properties of the set.

>>9078528
small IQ sjwtard detected.

leave

>>9079632
So let's look at a more general case:
Until now, S was defined only for coupls a,b that already satisfy R. But what if they're just any two elements from A? I.e - aSB iff f(a)Rg(b), no matter if aRb or not.
In that case, if I understand correctly, the only requirements that f and g must satisfy is that they map a and b into the same class of R inside A. Meaning, f: R(a) to R(n) and g: R(b) to R(n).

>>9079801
Yes
Also you should check out
Orbit-Stabilizer Theorem
Burnside's lemma
Polya Enumeration Theorem

Basically they are concerned with un-labeling the set that a structure is defined on.

The names you give to the elements of a set are arbitrary. abc 123

It is how the structure itself relates the elements of the set to each other that is important, not the element's "names".

>>9079858
Thanks so much!
I have absolutely no background in this topic, what exactly is it called? Is it just set theory or is it something else? Where can I read more about it?

>>9079955
Group Theory
Stick to finite groups starting off. (stay away from Lie groups and Lie algebras)
Learn the isomorphism theorems and all associated jargon.

Those specific topics revolve around Group Actions.

Honestly, chasing definitions on wikipedia works pretty well.

The un-labeling is "modding out" by the Permutation Group of the set (all of the ways to relabel the elements)