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 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.