Analyze Size of Pre-Image Sets Under Functions

[dumb_curiosity]

I am kind of curious what topics to read to understand this concept more.

Suppose I want to find a function [itex]f: A \rightarrow B[/itex], where if you look at the pre-image of any point [itex]b \in B[/itex], the size of the pre-image of b will be quite large. Essentially, I want to find functions that map from very large spaces, to very small spaces, with a pretty uniform distribution. In other words, I want to find functions so that if you take any point [itex]b \in B[/itex], you'll find that [itex]|f^{-1}(b)|[/itex] will be large (infinite hopefully), and pretty much the same regardless of which point you pick.

What are the "terms" that speak to the size of pre-images of points in a set under a function? What sort of area would I study to get a better grasp on functions like this, so that I might be able to "build one from scratch"? Topology?My apologizes for this oddly worded question, I just do not know what are the mathematical terms that express the things I'm looking for. The only thing that comes to mind that I could study is the kernal. I can look up tons of properties about the kernal of a set that would be useful. But the kernal is the set of points that map specifically to the null point. What would be the concept that is like a kernal, but for points going to any particular point, not just the null point?

 

[homeomorphic]

Too vague. If you can't find the right words, it would be better to talk about the context, instead of words that are going to be too ambiguous. What do you mean by size of the pre-image?

This could mean measure, in which case, it would be measure theory. Topology doesn't deal with size, but there are things like fiber bundles where each pre-image point is the same topologically and fibrations, which is a similar concept. But since topology doesn't deal with size, the pre-images could be vastly different in size--for example, one could be a massive triangle and one could be a tiny triangle.

In group theory, the kernel is the pre-image of 0, and the pre-image of another point is a coset of the kernel. In linear algebra terms, you'd just get some affine subspace that is a translation of the kernel (which would be an example of a coset with respect to the addition operation).

 

[dumb_curiosity]

By size I was talking about how many points are in the pre-image, not size in any literal way. (Just out of curiosity - what is a better term than "size" to use when talking about how many points are in a set?)

>pre-image of another point is a coset of the kernel

Oh I didn't realize this - I think this is the sort of thing I was hoping for. I know where to start now!

 

[homeomorphic]

what is a better term than "size" to use when talking about how many points are in a set?

Cardinality. For finite sets, you can also use the word size, but you said "hopefully" infinite, which means you need to specify whether you mean measure or cardinality (or what sort of infinity is it).

 

[dumb_curiosity]

Is there any notion of groups in which all cosets of the kernel are the same size?

 

[homeomorphic]

Is there any notion of groups in which all cosets of the kernel are the same size?

They are always the same size (cardinality) automatically.

 

[dumb_curiosity]

Oops - looks like I wrote my comment the moment you did. Thank you for pointing out the cardinality term, that is what I should be using. My "ideal" function would be one in which the carnality of the pre-image of all points (cosets of the kernel it seems) is inifinite. Mind you, it doesn't matter what type of infinite (sorry, again I don't know the term, what I mean is, I don't care if homeomorphic to R vs. Z. Just that the carnality would be infinite)

 

[homeomorphic]

The definition of kernal and cosets only apply to homomorphisms and not just any map between groups, by the way. So you just need a homomorphism that has infinite kernel.

 

[dumb_curiosity]

>They are always the same size (cardinality) automatically.

Maybe I'm misunderstanding things then. There are functions in which the pre-image of some points contain more points than the pre-image of other points. An example - I know it's trivial but a function f defined on R, so that f(x) = { 0 if x < 0, x if x >= 0

Here the pre-image of 0 has infinitely many points, while for any positive a, the pre-image is a single point.

 

[dumb_curiosity]

>The definition of kernal and cosets only apply to homomorphisms and not just any map between groups, by the way. So you just need a homomorphism that has infinite kernel

Oh. This clears things up a lot.