Suppose |$A$| = |$B$| and let $f: A \to B$ be a surjective function. Can we/ should we define the ratio of size of set $A$ : set $B$ wrt function $f$?

Question

I've not seen this idea anywhere before, but I'm not a set theory buff either.

Let $A$ and $B$ be sets with the same cardinality and let $f: A \to B$ be a surjective function. Suppose, for every $y \in B$, there are exactly $n \in \mathbb{N}$ members of $A$ such that $f$ maps these members of $A$ to $y$.

Then define the Rubinson-ratio of the size of set $A$ : set $B$ wrt function $f$ to be $n:1$.

...and perhaps we may say, wrt function $f$, $A$ is $n$ times Rubinson-larger than $B$.

We can probably extend this idea in a few ways, but let's leave it as is for now.

Is this a known idea in set theory? Else, may I claim the name for it?

Answer 1

This idea comes up frequently in combinatorics. It is called a $k$-to-$1$ correspondence. It allows you to relate the cardinality of the domain and range via the $k$ factor, which can allow for counting one of the sets more conveniently than it would otherwise be possible directly. Also, I don't think that people can "claim" names like that. One can be the first discoverer, but names must be accepted by the relevant community. On a more humorous note, see Stigler's law of eponomy.

As requested by the OP in the comments, here is a link to a reference that uses this term in the finite setting. (However, a k-to-1 correspondence is also applicable to an infinite setting where the two sets $A$ and $B$ share the same cardinality $\aleph_i$)

Answer 2

let's talk about the Rubinson ratio!

First of all, just to answer your question directly, I have never heard of a name for this concept in the context of infinite sets, although there are a few related concepts that I'll get into shortly.

But I've also never seen this concept used (other than as a part of the related concepts). There's an saying that "a mathematical object is what it does." What does this concept do? What sorts of proofs can we construct with it? You can call this the "Rubinson ratio," and you can say that that's the name of it, but until and unless you do something with it, it's not really yours. The person who gets to name a concept generally isn't the first person who comes up with it, but the first person to do something with it.

In my (albeit limited) encounters with set theory, it's been very rare that I'm using a function like you describe, between two sets of equal cardinality, where the size of the preimage of each element in the range is equal. However, I think you're essentially trying to define a way to think about the "size" of one infinite set compared to another, and in that case we do have a few examples:

First, if we have a subset $A\subseteq\mathbb{N}$ of the natural numbers, we can define the Natural density of $A$ as follows: First, we define $f_A(n)=\big|A\cap \mathbb{N}^{\leq n}\big|$ be the number of elements of $A$ that are $\leq n$. Then the natural density $d(A)=\lim_{n\to\infty}\frac{f_A(n)}n$. This works similarly to how you would expect: the density of the even numbers is $\frac12$, as is the density of the odd numbers. The density of the multiples of $3$ is $\frac13$, etc. The primes, the squares, and all finite sets have natural density $0$.

There's also a related concept in group theory: If we have a group $G$ and a subgroup $H$, then we can define $[G:H]$ (read as "G over H") to be the number of left cosets of $H$ in $G$. When $G$ and $H$ are finite, $[G:H]=\frac{|G|}{|H|}$. We also have $[\mathbb{Z}:2\mathbb{Z}]=2$ and more generally $[\mathbb{Z}:n\mathbb{Z}]=n$. In general, the function $f:G\to (G/H)$ which maps an element of $G$ to the coset to which it belongs maps exactly $|H|$ elements of $G$ to each coset.

There are other concepts like this in other fields (@Favst gives a good example from combinatorics), but I'm not aware of a general term that applies across fields of mathematics. If you ever publish a paper that uses this concept, try naming it the Rubinson ratio! Maybe it'll be in a textbook someday.