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... (read more)
I used your axiom list and Zorn's lemma proof sketch to make Mnemosyne cards. Thanks a bunch!
Thanks, habryka. I added a short explanation and linked this in the post. I thought it would be more common knowledge than it is around these parts.
In whales' excellent Book Review: How Learning Works, each one of the 7 Principles gets expanded upon with a "Research consensus". In the first principle, discussing how prior knowledge affects a students' learning, they write:
Declarative knowledge (object-level concepts) and procedural knowledge (how and when to apply those concepts) do not always go hand in hand. One without the other is a knowledge gap that can be... (read 885 more words →)
Nice dude!
I don't think this refutes my essential point, but it does add a caveat to it that might help exceptions realize when such a course wouldn't actually help them much. I've never taken a course in logic, and have in fact only recently cracked open a book on FOL proper.
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.