Why do we define functions to be set theoretic objects?

Question

Why do we define functions to be set theoretic objects? Functions are so intuitive, why do we define it in complicated set theory language?

Answer 1

You have not necessarily to think in term of "definition" as a sort of "replacement" of the intuitive notion of function with its set-theoretic counterpart.

We can say instead that set theory provides a "model" for the mathematical concept of function.

Functions was already known in mathematics well before set theory. With the set-theoretic definition of function as a set of ordered pairs we have a simple and very useful "model" for them which, of course, does not contradict the "usual" behaviour of functions in mathematics.

See also : Function

Answer 2

We do it like that because we have a very good, rigorous grasp on what a set is, so we can speak of a set of pairs representing a function with no ambiguity at all. This is precisely the reason why we often construct everything out of sets - the integers, rationals, reals, and so on. This is mundane, and not very enlightening, but if you ever have a doubt if, say, $1/2$ and $2/4$ are the same fraction, or if $0.999...$ and $1.000$ are the same reals, you have a rigorous way of answering this question.

On the other hand, we all know what that a function $f\colon X \to Y$ is "some sort of rule" for assigning a value $f(x)$ to any $x$. But try to give a rigorous definition to "some sort of rule" in such a way that it won't be possible to misunderstand it...

Answer 3

Functions are only intuitive if you think about $f(x)=x^2+1$ or $f(x,y)=\ln x+e^y$ or so on. But how do you describe in an intuitive manner every function from $\Bbb R$ to $\Bbb R$? There are more than you can possibly imagine. How would you describe intuitively a function between two sets which you can't describe? There are sets which are neither intuitive, nor obvious. And one would expect that some functions would have these sets as domains.

Intuition can be misleading, and intuitive objects without a formal definition may cause mistakes. Look at the history of the definition of a function. It was believed that all functions are piecewise continuous, but that wasn't true; and that all continuous functions are differentiable almost everywhere, but that's not true either; and as time progressed we learned that in fact the "intuitive" part of the functions make up but a minute and negligible part of all things which are functions. (The reason is that intuition changes between people, and from time to time; and there is a slippery slope here: if the existence of this object is intuitively clear, then that object must exists, and so on, until you get somewhere that you have no intuition about.)

So instead, we have a formal definition of a function. And then there are less mistakes.

Of course one doesn't have to take a set theoretic definition of a function. This just one way to model the notion of a function using sets. One can use other means to do it. Using sets however, one can model almost all mathematical things, and the fact that one can model functions using sets is important for that.