| I have no problem with using notation in a more 'intuitive' way than the technical definitions allow. However I won't ever acknowledge such notation as correct. If you want to be precise you'll need to use precise notation, and insisting that abusing the '=' sign is precise will do more harm than good. Unless you consider 1+1=3 to be good notation. If you seriously think you're not missing out by removing the distinction between ⊆ and =, consider that O and subset relations are enough to denote all relations that you'd normally use the different big Os for that nobody ever bothers to remember. So f being of order o(g) is equivalent to O(f) being a proper subset of O(g) (O(f)⊊O(g)), f being of order Theta(g) is simply equivalent to O(f)=O(g), and f being of order Omega(g) is simply O(f)⊋(g). You might need to restrict yourself to nonnegative monotonically increasing functions, but that's pretty much done in practice anyway. And this might differ from some of the existing definitions that are being used, but those do vary a bit across sources, and these definitions are as sensible as any, and are compatible with the natural partial order on sets of the form O(f) given by inclusion. Seriously though there's nothing you'll gain from throwing away a nice inequality relation just because it saves you from writing ⊆. I'm also not sure what you have against thinking of something as sets, but you can't be precise in mathematics and insist you're not using sets, at least not without going through a lot more trouble then you'd ever avoid that way. In particular it's fine to think of: >“when you take n and add a quantity that is at most a constant times log n, and square it, you get n^2 plus at most a constant times n log n” and >“the set of functions obtainable by adding the function n ↦ n to a member of the set of functions that map n to at most a constant times log n, and squaring the sum, is a subset of the set of functions obtainable by adding the function n ↦ n^2 to a member of the set of the functions that map n to at most a constant times n log n”, as being the same thing. One is just written in slightly more obnoxious math speak. However if you insist that the meanings are different then something is wrong, because that would imply that the most straightforward mathematical interpretation is apparently different from what you meant. |