The inverses Halmos defines here are more general than the inverse functions described on wikipedia. Halmos’ inverses work even when the functions are not bijective.

I believe that what you are speaking of here is Halmos’s discourse on what are called these days “images and preimages” or “inverse images”. I found the subtle difference between these and inverse functions proper annoying when I was learning proof writing, so let me illustrate the concept, so that we have a *caveat emptor* for the budding mathematician.

Take the sets A = {0, 1} and B = {2}, and define a function f: A → B as f(x) = 2 for whatever x in A you throw in there.

Then,

f(0) = 2, of course.

f(1) = 2, as well.

f(A) = {2}, which is the

*image*of the whole*set*A “under” the function f.f^{-1}(B) = {0, 1}, which is the

*pre-image*of the whole*set*of B under f. Meaning, “anything I can throw into f, from A, to get something in B”.f^{-1}(2) , however, is

**meaningless,**at least as far as functions go. Functions can only return one thing, so how would you decide whether f^{-1}(2) should give you back 0 or 1?

If you say f^{-1}(2) should give back both, well, now you’re not dealing with an inverse *function* any more, you’re dealing with the inverse *relation*. You can in fact deal with that, with some other tools in the book

These are more general, which is nice, but I’ve found that in a rigorous environment it won’t do to describe them with the same language you use with functions. You really want to toy with these gently, if you can.

I quite like this approach. :) I’ll see if I can apply it to electrical engineering and pure mathematics soon, as those are the subjects I am studying in school. Linear algebra will be my first stop.