division

[phoenixthoth]

trying to define n/m when it exists in N.

suppose m and n are natural numbers and let [n,m] denote all functions from n (which is {0=Ø, 1, ..., n-1}) onto m. if [n,m] is empty, stop and say that m does not divide n.

consider the subset of functions f in [n,m] such that ~ defines an equivalence relation on n where x~y iff f(x)=f(y) and that each equivalence class has the same size q. call this set [[n,m]]. i'm guessing that if [[n,m]] is nonempty then it will be the same q for all functions in [n,m] such that ~ defines an equivalence relation partitioning n into equal sized parts (m parts each having q elements).

i suppose this is equivalent to saying
[[n,m]]={f in [n,m] : for all z in m, card(f-1({z})) is constant}.

definition: if [[n,m]] is nonempty then let n/m=q. if [[n,m]] is empty say that m does not divide n.

question: n/m=q in this sense if and only if n/m=q in the usual sense? (i'll also have to decide if q is well defined.)

this is a definition of division not obviously equivalent to "the inverse of multiplication." one can now define multiplication to be the inverse of division!

[Hurkyl]

All right, let me see if I understand you.

In order to lessen potential sources of confusion, I'm going to use the notation \mathbb{N}_n = \{0, 1, \ldots, n-1\}.

Given two natural numbers n and m, consider the set (\mathbb{N}_m)^n \; (\,= (\mathbb{N}_m)^{\mathbb{N}_n}\,), the set of all functions from \mathbb{N}_n[/tex] to [itex]\mathbb{N}_m.


We can then define n/m := q where q is a number having the property that there exists a function f such that \forall z \in \mathbb{N}_m : |f^{-1}(z)| = q.


This is well-defined; q exists iff n/m exists, and q is unique when it exists, but I fear that you might need to know multiplication to prove the uniqueness of q.


Edit: fixed logical mistake.

[phoenixthoth]

yeah, that's what i meant though my (\mathbb{N}_m)^n is the set of such functions that are onto.



I fear that you might need to know multiplication to prove the uniqueness of q.

that would suck.[g)]

i'm really aiming and doing a similar thing for infinite cardinals...

[Hurkyl]

There's a problem for infinite cardinals:

Consider the following elements of \mathbb{N}^\mathbb{N}: f(x) = x, g(x) = \lfloor \frac{x}{2} \rfloor, and h(x) = n - 1 where p_n is the smallest prime dividing x + 2.

f(x) partitions \mathbb{N} into the equivalence classes 0 | 1 | 2 | 3 | \ldots, indicating that \frac{\mathbb{N}}{\mathbb{N}}=1.

g(x) partitions \mathbb{N} into the equivalence classes 0, 1 | 2, 3 | 4, 5 | 6, 7 | \ldots, indicating that \frac{\mathbb{N}}{\mathbb{N}}=2.

h(x) partitions \mathbb{N} into the equivalence classes:


0, 2, 4, \ldots | 1, 7, 13, \ldots | 3, 23, 33, \ldots | \ldots
[/itex]

indicating that indicating that \frac{\mathbb{N}}{\mathbb{N}}=\mathbb{N}.


Notice, though, that

[tex]\mathbb{N} = \mathbb{N} * 1 = \mathbb{N} * 2 = \mathbb{N} * \mathbb{N}.

[phoenixthoth]

this concept of "division" seemed to capture the statement \mathbb{N} = \mathbb{N} * 1 = \mathbb{N} * 2 = \mathbb{N} * \mathbb{N}
. i suppose no definition of division would be well defined for infinite cardinals.

[Hurkyl]

though my ... functions ... are onto.

This is a superfluous condition if you do it right; I didn't do it right when I first wrote the post. [:)] If a function isn't onto, then the inverse image of some element will be the empty set (which is not q), so we don't have to restrict ourselves to onto functions.

[phoenixthoth]

yeah i was wondering if you omitted that because you realized it was superfluous.

hmm... i wonder about 0/0

it would seem to be the case that by my definition, 0/0=0.

i also wonder about P(N)/N.