matt grime

You can take any set and tak it modulo something. The result is a set of sets of equivalence classes. It may happen to be that that original set is a product of two sets, but the result is not a modulo set product, it is the product of sets modulo something, and the "product" has nothing a priori to do with the "modulo" just as it has nothing to do with the product in modulo arithmetic! So I can't give you what you're after because it doesn't exist. At the risk of repeating myself, when you do [2]*[3]=[1] mod(5) you are not taking the cartesian product of the sets of equivalence classes of 2 and 3 and equating them. modding out by an equivalence relation happens all the time. You may do something like it in the ring theory course if you were to say take the ring of polynomials in one variable and mod out by an ideal.

You can and should always write 0 for the additive identity in any ring. I don't think you're going in a the right direction at all. The product of arbitrary sets has nothing to do with multiplication really. Moreover, multiplication of integers modulo some number does not in itself define any equivalence classes as your last post seems to imply you think. The equivalence classes are defined first, then we show that these equivalence classses inherit the structure of a ring where the * comes from the * in the integers. We are not actually taking any _set_ products at all. You do this a lot in mathematics, and you'll often hear us say two things are equal modulo a third as a short hand of saying two things are the same up to their differences, where their differences aren't important. For instance, suppose we try to solve a set of simultaneous equations in matric form Ax=b, and suppose that A is singular, and that there are many solutions. Let u and v be two solutions, then A(u-v)=b-b=0 and we might say that any two solutions are equivalent modulo the "kernel" of A. (The kernel being the set of solutions to Ax=0). You know about linear algebra right? Well, you can show that the kernel of A is a vector space, and that modulo the kernel all answers are equivalent.

The first relation merely states they are orthogonal. It is not necessary nor for a complete set to be orthogonal, nor is it necessarily true that an orthogonal set is complete. In particular that relation remains true if I allow only one eigenfunction in the set. I don't understand what the second equation states, really, cos I'm not a quantum physicist. Whether the eigenfunctions are complete is a property of the linear operator of which they are eigenfunctions. (the eigenfunctions of any hermitian operator are orthogonal and real, though not necessarily complete)

Indeed it doesn't since you've not defined what you think + means for sets. There is the idea of Union of sets. What do you know about Rings? Sets form a ring under the operations union and symmetric difference' date=' but union is actually the multiplication operation. and symmetric difference the addition. Firstly that isn't AxB, that is some subset of AxB. And if you didnt' want the b's to all be equal then you should have used different letters. The set you've written down clearly has cardinality k, not k*card(B) as would be required. Also, sets aren't ordered. There is no "first" element of a set in any canonical sense. As i keep saying, one doesn't say the "modulo multiplication", one says "multiplication modulo". the modulo comes aftewards if you like. you've not defined modulo set product so how can you talk about it as if you had? If I remind you of a prof there may be some reason for that, though I am not a professor. You're hung up on multiplying sets as if they're numbers. They aren't numbers. Just because two things have a similar name doesn't mean there is anything more than a formal similarity between them. I can take two sets, take their product, and then take the "mod an equivalence relation", I can take them "mod a relation" anyway. What is important in modulo arithmetic (forgetting the set stuff) is that taking the equivalence classes preserves the algebraic operations and is independent of the choice of representative of the equivalence class. That is [a]=[ab] In order for this to make sense in terms of sets in a direct correlation I would need a multiplication of sets, and an addition of sets to preserve under the relation. But set product isn't it! the product of sets isn't a multiplication of sets in the algebraic sense: it has no identity element. So, you're mixing up a couple of ideas here.

If the sets are finite.

To whet your appetite some more, how about this? http://www.dpmms.cam.ac.uk/~wtg10/commutative.html

http://www.dpmms.cam.ac.uk/~wtg10/ordinals.html there's a page with some uses of transfinite induction to prove statements you already know (and some you won't). Note, these proofs are pedagigical in nature. The usual proofs are much easier to understand.

Semantically: induction does not prove anything about the natural numbers necessarily. It is used to prove things about any well ordered (countable) set of statements. Usually that list is indexed explicitly and obviously by the natural numbers, but the result doesn't have to say anything about natural numbers. As it is any countably infinite well ordered set that you need to use at this stage is canonically isomporphic, as an ordinal, to the natural numbers, which is why it almost always appears to use the natural numbers. The negatives. with the obvious ordering are isomorphic to N with its ordering via x sent to -x. Infinite sets of greater cardinality may indeed be used to prove statements inductively. This is so called transfinite induction, and is most inelegant, and would require too much of an explanation of uncountable ordinals to be of any use at the level where you're first using induction. Indeed I can't think of any place where you'd need to use it before graduate level mathematics.

And I thnk it is valid to ask why you think there ought to be a modular set product. What relation between the product of integers and the product of sets do you think exists? (There is one, but I want to know what you think). Also, no one refers to 2*3 = 1 mod 5 as being the modular product of 2 and 3, they would say the product of 2 and 3 mod 5 is, and you can mod out anything by anything pretty much if you felt like it. For instance the product of the circle S^1 with the unit interval [0,1] modulo the relation (x,0) ~ (y,0) for all x,y in S^1 and (x,1)~(y,1) for all x,y in S^1 is essentially the same as the unit sphere in R^3.

A function is a subset of the product XxY but certainly isn't all XxY. A product is not a function. You've still not given me any way of understanding what it is you think that ordinary multiplication has to do with set products that makes this generalization reasonable. Ie what are you trying to abstract to the modulo case. in modulo arithmetic when you add together two elements x+y you need to demonstrate the answer is independent of whichever choice of representative of the equivalence class you choose. What if X and Y weren't sets of integers. what would you want XmodxY to even mean?

I don't think that it is true to say that the y in (x,y) is dependent on x. Firstly, I don't understand what you mean by dependent on in this context. The product of two sets is defined for all sets, and is the collection of all ordered pairs of elements, so you give me (?,y) and I can't tell you what ? is except that it is some element of X, and moreover that such a pair exists for every ? in X. You appear to be getting at partitions of sets. Which is equivalence relations. One such equivalence relation is modulo arthmetic. You're just partitioning a set, it's not called modulo product as modulo stuff is a special example of this kind of behaviour. What I thought you might be getting at is if X and Y are two finite sets of cardinality p and q respectively, then the cardinality of XxY is the product of p and q.

I once had to read a book where typically "the proof of Proposition 4.5.6" was: "this follows from 2.1.2 and 2.4.5 and definition 1.2.1 and corollary 1.6.7" some proofs even required external references to be checked, which is acceptable in papers, but unnecessary in textbooks if the proof is essential.

By names if they have them. Examples were given: Hahn-Banach Fundamental Theorem of Calculus/Algebra/Arithmetic 3-epsilon or epsilon over 3 Morera's Theorem (for triangles) Cauchy's Theorem (there are many of these) Green's Theorem Stoke's Theorem and the list goes on. Those unnamed are described, even if you're writing a thesis/paper and you number something as theorem 3.3.1 it is *BAD* mathematical presentation to, at a later point, simply say "this follows from 3.3.1", and infinitely preferable to say "this follows from 3.3.1 which showed that ...." Not everyone has a photographic memory.

What relation do you think "modular set products" ought ot have to modular arithmetic? I mean are you trying to generalize a perceived relationship between set products and ordinary arithemetic, and if so what relationship is that? (This isn't vacuous, by the way, but I'd like to see what you think you're getting at)

Some axioms are numbered and reasonably well known. The axioms of (euclidean) geometry for instance. And the seperation properties of topological space (T1, T2, T3 etc). Not to mention that triangulated and other similar categories have reasaonabl well known numberings of axioms (AB1-4).

Is it easier to catalogue the millions upon millions of theorems and have a handy reference manual we all can ue (how would we all use it?) than to simply state the required result? Do you know how many theorems there are in the maths world? The important ones have names that are familiar to most, and that is sufficient.

And he's not even demonstrated that there is a model of R in Monadic Maths either. Or in fact that there is anything that is a model of anything that exists in a model of MM, never mind anything as complicated as the real numbers, or addition.

What is the order of xy if x,y are elements of finite abelian group with ord(x)= p and ord(y)=q?

You appear not to have explained yourself very clearly. Are you saying that yuo are defining f(z) for z in R to be the limit of the sequence x_{n+1} = (x_n)^y+c with x_0=0 or what? what is the index over which you're iterating? x,n,y,m you've used all of them and swapped the use of n in the posts too. So try explaining it more clearly.

Your conjecture is doesn't depend on n or C, are you sure that's what you want? If n>1 If c=0 the 'function' obtained by iterating repeatedly connverges pointwise to zero for x between -1 and 1, is 1 for x=1 and =/-1 for x=-1 and diverges for all other real x. if c is greater than 1 it divereges for all positive x, and if n is even diverges for all x. If n=1 it diverges for all x, htink abuot the cases a bit more.

For some reason I seem to remember reading that the military (UK) used grads for artillery finding 400 more practical for people to work with in their heads: 100 per quadrant, 50 for, erm, octant???

Well, I'm at neither place, and I would put my current employers inbetween both in terms of quality of research. The quality of undergraduate 'ought' to be better at Warwick, though that doesn't mean any single person is better than any other single person irrespective of their flexibility, since Warwick now screens using STEP doesn't it? And Nottingham doesn't. Nottingham is still a good university.

of course it could, since these are just notations for objects with a formal set of properties. we could have given them any meaning we wanted. in fact in base 11, 0.999... doesn't equal 1. the thing about being very close to it indicates that you want a system where the archimidean axiom is not true, that isn't a very useful system really but you may define it, and use it. note that in that system you cannot prove almost any theorem that you take for granted in analysis (even limits are no longer unique).