Oh, I think in English notation it would be | instead of :

As for the first I have also another idea,
E={e | ∀e, e' R(e)=R(e')}, but that just looks ugly af

E!xFx, where F is a predicate and E! is pronounce "E shriek" according to Bertrand Russell. Quine uses an inverted iota for "E!" but it's the same idea. For the one and only F. Sometimes defined Ex(Fx & Ay(Fy iff y = x), where E and A would normally be inverted and iff is a triple bar.

What i don't like is iff seems to be rendered as '⇔' in unicode, but other than looking at that, i can't figure out how to help...

woah, I honestly understood so little of what you wrote : D
I'm gonna investigate though

Russell wrote about "the so and so" in Principia Mathematica (with Whitehead). I think there's a paperback with just the initial part. Quine wrote about it in Mathematical Logic and elsewhere. It's usually discussed soon after the notion of identity is introduced in set theory or in mathematical logic. My references are very old. I'm sure there are more contemporary ones.

How about something far simpler:

f (x) = c | c ∈R

Also, on the second one, if you're going for a binary set, it looks alright, although we usually note it as

S={s1, s2, . . . sk} instead of m. K is more regarded as an undefined integer constant (m is as well, although we use k first).

Also, steer clear of e in your post unless you're directly referring to the identity element.

Tru Ninja, as for the second one, the m is inherited from the number of elements in an other set. For the same reason it's j not i.
So there is the vector S with coordinates equal 0 or 1, and c_j is included in the A set iif the j-th coordinate of S is equal to 1. That'd be A={c_j∈C : s_j=1}, right?

As for the "e", it's actually E as an element of the squiggly E (apparently doesn't work in unicode)
http://screenshooter.net/102518457/otivvnf

I see what you're referring to.

My expertise ends at group and ring theory, although where that's concerned, if you have a set with one element, such an element must be the identity and e=e' is given.

If you're referring to anything outside algebraic theory, then you'll have to probably be a bit more specific for what you're looking for. In that case, this makes the most sense:

E={e | ∀e, e' R(e)=R(e')}

What is it that you want to define? Tru_Ninja's formula doesn't make sense since e doesn't occur free in what follows "|". Do you want the set of *functions* what have one and only one value for any given argument?

Ok, so with the context:
e is an election, R is a social choice function, that for given e returns a set of winners. I need a set of elections - E, such that for all e in E R(e) is the same set of winners.
I'm actually defining a class K=(E,R) here, but I don't think it changes anything, does it?

Also, since I already got your attention, I cannot find a trustworthy answer:
If I'm only interested in the cardinality of a set, dunno, e.g. {x∈N | x<10} could I just note that as |x∈N | x<10| or would it be |{x∈N | x<10}|?

Since you want the cardinality of a set, I'd think the 2nd unless the 1st is somehow understood as shorthand for the 2nd. Actually, the 1st is confusing since it has three |'s.

So you're essentially defining a set of sets:

E = {e1, e2, . . . E sub n | e sub i = { R1, R2, . . . R sub n} }.



Is this for the purpose of programming?

yeah, but actually I'm gonna use : instead of the middle | anyways

Or more thoroughly

For all e, an element of E, e sub i = R1, R2, . . . R sub n.

Using that in place of the equals in the second part.

That's the written notation I'd use. To swap it to a program, I'd defer to someone more fluent than myself.

Hah, actually I thought about including some programming examples to show that the computational complexity of the problem is not an issue, but decided it would make the paper too long. It's a part of follow up to my thesis.

yassem,

As asked, the set you're asking for is not well defined. You want the set of e such that R(e) is the same AS WHAT? There are many different sets for which the value of R(e) is all the same -- they're called the level sets of R.

Let me give an example. Consider the function f(x) = x^2. What if I say I want to define the set E where the value of f is always the same? Well, there are many such sets. For example, {-1,1}, {-2,2}, etc. Anyway, there IS a notation for any one of these sets. Let's say I want the level set that are all the same as 1. Then the notation would be

f^(-1)({f(1})

or the inverse image of {f(1)}. In your example, if you want all the ones that are the same as e' (where e' is some fixed election), then it would be

R^(-1)({f(e)}).

Or if the result is known, say R', then it would be

R^(-1)({R'}).

By the way, there isn't really a parenthesis around the -1 -- it's wrtten as an exponent, just like the one in inverse function notation, f^(-1).

Actually, the fact that there are many sets of such elections is the whole point. Ok, so a bit more context (I guess it will be beneficial for me as well to talk about this, get some expertise before my thesis exam):
I am considering a consensus class. A consensus is a situation in which the outcome of the election is so obvious it's bordering on "natural". Example: weak unanimity. So you've got the social choice function R - in this case it would be choosing the candidate who is preferred the most by all the voters. Then, you consider the set of all election, all possible elections, all sets of candidates, all possible preference profiles, all. And then you build so many sets of E as there are possible winners according to rule, in such way that in one E for all elements e the result is the same. In that way I receive my consensus class K=(E,R).

I honestly hate this situation, the paper I am basing this on just says "...for which the result is the same", but my coordinator tells me to use mathematical notation.

Wait, I've actually got an idea, but first: would you consider {e ∈ E | ∃R(e)} a correct notation? There are many cases in which R(e) wouldn't give a value, so I need a set of elections for which it would return a value.

This conversation would be much easier to have using mathurl:

http://mathurl.com/obohbbs

This site is so fucking awesome : O
http://mathurl.com/ofcnory

Of course in the definition of sets : is meant to be | in English notation, apparently.

Are you trying to define A as a set or a sequence? If A is a set, then you can't say the j-th element of A, since sets are unordered.

No yassem, that is not good notation.

What are you wanting to index the different sets E by? That is really the question. For example, you might define

E_e = {f : R(f) = R(e)}

Then every f would be in some E_e (in particular, it would be in E_f).

>inb4 Zultar bans you for asking for help with schoolwork

Well, the elements of A are unordered. c originally is an element of C, which is also a set - unordered. (by the way - c is a candidate) Nevertheless, for the sake of notation, in some cases I need to be able to specify a lot of elements of this... and oh fuck I just now realised that C is in fact ordered. I mean, I do need to have c_1 c_2 and so on, because the coordinates need to refer to those elements. But the original paper still referred to it as a set... It is a sequence as fuck. They even say that it doesn't matter what order the elements are in, but if it were a set, there wouldn't be any order, would there be? Damn, I trusted those ppl : D

@Semck, I don't need to index the sets. They just need to exist. At one point I'd for example say "the distance between the election and the unanimity consensus class in which c_1 is the winner", but before that I need to define what a consensus class is. But now that I started doubting the paper, I actually think that I do not need to include the part about all the outcomes being the same. It doesn't matter for the sake of the distance.

This seems more legit than asking for a dataset to do some random OLS on a few hours the assignment is due.

I asked for an inspiration, not dataset : D
Got a 4.5 out of 5 BTW, worked out pretty great : )

Well, the point is, you *do* need to index them if you want to refer to them.

Yeah, but I'm gonna take a whole lot of heat because of that. I mean, it is obviously logical. You even have criteria that explicitly state that the order doesn't matter, and most voting rules fulfill those criteria - so by the most common notion it's referred to as a set. Everywhere there's "set of candidates". But then, in so many cases they refer to c_1, c_2 and so on. How could the coordinates possibly be assigned to particular candidates if they weren't ordered in a sequence, right? Damn, that is gonna take some serious detail changing in my thesis...

Oh, you referred to the E's. I really don't. Trust me on this one. There are infinitely many sets in that class. Uncountably many. I could index them by the set of winners they return under given consensus voting rule, yeah, so I could have just another index and say that all R(e) equal that index, but I really do not want to index them, because then I have a big problem later on.

But then I have another problem. Voter's preferences are supposed to be a linear strict order over set C. You cannot make an order over a sequence, only over a set, right? I could say it's a permutation or the said sequence, but then I won't be able to discuss cases in which the order isn't strict... Damn this is problematic.

Nope, I'm just gonna go with the official notion. I'm just gonna ignore the fact that this is obviously a sequence and call it a set nevertheless. Considering it a sequence would be a nightmare. No surprise no-one does that : D

And you know what, fuck it, it is a set. The elements are just indexed. http://mathurl.com/nhznxfu

Then how about just say, "We call a set E an *equivalence set* [or whatever] if it is a maximal subset of the set of rules such that the outcome is the same for all members."

OMFG, the problems I encounter start make me really start hating mathematical notation. I don't suppose there is a way to say "Sequence V with and additional element x". Such that, by denoting it, dunno, "+" you'd have (1,2,3)+{4}=(4,1,2,3) v (1,4,2,3) v ... and so on.

@yassem, listen to semck. He's right.

I personally would write
E_r = {f : R(f) = r},
where r indexes all the potential results of the election.

I think you do need a sequence rather than a set since the same person can run for more than one office. At least in the US. It has happened that someone has both run for Senate and President at the same time. I think.

Ok, so since the last time I asked you some questions I got some quite helpful answers I'm gonna exploit you some more. This time though, learning on the previous mistakes I'm gonna just include the background. Also, I think it'll be just easier if I go ahead and write it all in mathurl.

http://mathurl.com/o3nsuf3

i am still very bad at LaTeX, but I discovered I can write in something similar to latex in word, which is cool.

Anyway, thanks in advance for your help.