Okay, so I've gathered that compositions are typically defined by initializing functions with codomains in mind, where in general, the range of g, given the range of f as input, is a smaller set than the codomain that we used to initially define g. The codomain of f (by which I mean the initial...
Interesting. I thought from the way he seemed to use those words that "codomain" and "range" were the same idea. I guess I'll have to sort through that too. He does in fact use "range" in his definition. This is the first time that I'm hearing those are different ideas. Thanks for letting me know.
It seems that Tao defines compositions in a different way. He defines compositions so that the range of f is precisely the domain of g. Maybe he does this because it's more natural from a foundational point of view at this point in his text and maybe it's too restrictive in general;. I'm not...
Using that more general definition of composition only helps me if its true that functions have a unique domain, because you refer to "the domain of the outer function". Are rules for mapping between sets assigned their own unique domains? This general definition of composition just seems to...
I just finished working through compositions of functions, and what properties the inner and outer functions need to have in order for the whole composition to be injective or surjective. I checked Wikipedia just to make sure I'm right in thinking that for a composition to be injective or...