# Andrew Quinn

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• 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.

• 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.

• 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.

# Two pre­scrip­tions for fix­ing a pro­ce­du­ral/​declar­a­tive knowl­edge mis­match.

22 Jun 2018 22:17 UTC
35 points