[Updated] Need help checking set theory mathematical notations

[MobiusPizza]

I am a beginner in set theory and more importantly the notation conventions. I am trying to describe what I am doing below in mathematical notation:

Given a set A, containing elements which come in non-distinct pairs e.g. {a,a,b,b,c,c,c,c,d,d,.....}
I am to partition A such that
P1...Pn, being subsets of A, contains identical elements. For example, P1 = {a,a}, P2 = {c,c,c,c}, etc.

I can use plain English but it doesn't sound cool

Thank you very much

 

[iCyborg]

The {list_of_elements} notation is quite standard. The only thing that's not standard is referring to a collection that contains duplicate elements as a set. The word you're looking for is multiset.

 

[MobiusPizza]

Thanks, I found out multiset and classes (collection of sets)
I was able to write the following:

I just need some Mathematics guru to check over the syntax. q and r are defined constants. R^q is the cartesian product of R q times.
I mostly followed the ISO-80000-2 notation convention. I used {{}} to indicate multisets, <> to indicate classes.

Below is still valid:

Given a set A, containing elements which come in non-distinct pairs e.g. {a,a,b,b,c,c,c,c,d,d,.....}
I am to partition A such that
P1...Pn, being subsets of A, contains identical elements. For example, P1 = {a,a}, P2 = {c,c,c,c}, etc.

Except set A is now really multiset S, and P is a class which contains all possible S. The partitions are now Q (= P1, P2 in the old quote), which should contain the largest partition of common elements only. In the quoted example {c,c} is not valid cadidate for Q while {c,c,c,c} is. The union add symbol in the equation is to simulate this partitioning behavior.
Lastly, the elements are acceptable to Q if there a even number of them, except if the elements are numerically equal to r, a predefined constant, then there must be odd number of r's.

It still looks awfully complicated to me, any simplifications?

Thanks again in advance

 

[iCyborg]

Your f(Q) doesn't make sense. E.g. if r is not in Q, it's defined as f(Q) = 2n for all n?? It should be some n, though if I understand your intention, you should just use something like
2 | f(Q)+1r(Q), | is fairly standard notation for "divides".

Also, you're using logical conjunction everywhere (the upside-down v), which means that r must be in every S part of P, but you seemed to imply it doesn't have to be?
I don't understand that 2nd conjunction and the union-add symbol...

 

[MobiusPizza]

Thank you very much for the reply, very much appreciated iCyborg

Your f(Q) doesn't make sense. E.g. if r is not in Q, it's defined as f(Q) = 2n for all n?? It should be some n, though if I understand your intention, you should just use something like
2 | f(Q)+1r(Q), | is fairly standard notation for "divides".

Oh thanks, basically cardinality has to be an odd if r is in Q

Also, you're using logical conjunction everywhere (the upside-down v), which means that r must be in every S part of P, but you seemed to imply it doesn't have to be?
I don't understand that 2nd conjunction and the union-add symbol...

At least one r indeed must be in every S, this is one of the criterion for forming S. S is basically one or more Qs joint (union-add) together. Put it another way, Q are partitions of S. Q may be just several r's (odd number of them) or some other number (even number of them). For a given number, say 1.5, Q is made up of the greatest number of 1.5 while still a subset of S, say four 1.5 units: {1.5,1.5,1.5,1.5}=Z are in S. The union-add symbol is there such that subset of Z are not contained in Q, for eample. {1.5, 1.5} can't be Q since {1.5, 1.5} union-add {1.5, 1.5} is Z hence is part of S. Therefore if Q contains 1.5 it must be {1.5,1.5,1.5,1.5} and no less.

An example is easier to understand:

If S = {1,1,2,2,3,3,3,3,4,4,4,4,4,4,r,r,r)
Hence Q = {1,1} ; {2,2} ; {3,3,3,3} ,{4,4,4,4,4,4} ,{r,r,r}
Q cannot take the form {3,3} nor {4,4} nor {4,4,4,4}
It's as simple as that. However this example is top-down, the intention is to say S is unknown and we are building S from Qs which satisfy the criteria set in the set-builder notation. From the all possible combinations of S we build P.

 

[iCyborg]

Ok, I think I got it now.

I'm not sure that {{R_q}} notation is proper, at least from wikipedia because that ISO document costs 140CHF
And it seems more like a multiset of q-tuples of R to me, although q has no meaning at that time, which is a problem on its own.

That 2nd conjunction still confuses me. Basically it reads 'for all Q<S such that (something)'. This doesn't really evaluate to true or false. It should be more like 'for each element there's a Q that contains it and has that condition satisfied'.
I'm not sure about the condition either: say Q={3,3,3,3}. There's no D={{x,x}} such that Q\D=0.


I think it would be easier to use multiplicity function:
http://en.wikipedia.org/wiki/Multiset#Multiplicity_function

You could say that 'for all x elt of S:
a) 2 | 1_S(x) + 1 if x=r (or use doesn't divide symbol)
b) 2 | 1_S(x)

Then you don't need all these Qs and weird contraptions.
Building these Qs in any practical way (whether by hand or programmatically) is pretty trivial anyway.