In lesson 6 we introduced the notion of a function, which gives us a way of relating two sets to each other. We defined a function so that it took every element in its domain to some element in its codomain, and moreover a function cannot send an element in its domain to more than one element in its codomain.
We made this definition relatively arbitrarily, meaning we simply said “here’s the definition, deal with it”. It turns out that this definition ends up being one of the more interesting ones, as it lets us ask all kinds of interesting questions. In other words, there are tons of different definitions we could have made for “relating sets to each other”, but this one happens to be one of the most useful. This was not by accident, as mathematicians have had a long time to study these ideas and realize which definitions are more interesting than others.
This is not to say that there are no other definitions that are interesting, however. There are tons of things that we can define that “relate sets to each other”, and several of these different definitions will be important and/or interesting in different ways. Moreover, every different definition will have different types of questions that can be asked about it, and different ways of “adding structure” to them. For example, with our definition of a function we were able to define, in lesson 7, the notions of injectivity, surjectivity, and bijectivity. These are special cases of functions, as we’ve defined them.
For this interlude, try to think of some other ways that we could “relate sets to each other”, and once you’ve made that definition, try to think of some various “special cases”. For example, I could relax the requirement that our function takes every element in its domain to something in its codomain, thus expanding the class of things we call “functions”. One special case of this new definition, however, would be the set of “functions” that do take every element in their domain to something in their codomain. We could call these functions, for example, “standard functions” (just a definition, based on this new “relaxed definition”). Then we would carry on studying these new “standard functions” just as we are studying our functions now, only we’d have this other class of non-“standard” functions that are not encapsulated in our current definition of functions.
For example, if we let and and if we define a “function” (with our new definition) from A to B as follows: , then this would not be a “standard function”. I.e., it does not satisfy our original definition of a function (since it does not send 1 in A anywhere), even though it is a perfectly well-defined object. Thus, our definition of a function simply ignores these objects, because for whatever reasons many mathematicians don’t believe they’re that important or interesting to study. Nevertheless, we could make that definition if we wanted to!
I therefore challenge you to think up some other ways of “relating two sets to each other”, and asking some questions about this new definition. For example, you could make the definition and then define some “special cases” of the definition the way that an injective function is a “special case” of a regular function (note that now, and forever, when I say “function” I mean our original definition—I don’t want this to get confused).
In the following link I’ll post some of my own ideas, so don’t look at them unless you’re looking for some motivation. In other words, I don’t want to steal your ideas before you have them, so don’t look until you’re okay with that possibility!