These days I was talking with some of my colleagues about our exams, and of course at some point we came to the discussion ‘How did you do exercise X?’. After a long debate about some metric spaces, I discovered that did not completely understood what a metric is, so I went to the proper definition that our teacher gave us, and from that I understood that they did not understand what a function is. I was a little shocked, but then I realized that we are using them so much, that we forget the definition and its true meaning. We just presume we understand it without thinking too much.
I will start by giving the definition of a function we received in our ‘Sets and Algebraic Structure’ course:
Definition 1.6: A function f from a set X to a set Y (written f : X->Y ) assigns to each element x of X a uniquely determined element f(x) of Y . A function is allowed to send different elements of X to the same element of Y , and there may be elements of Y to which no element of X is assigned. Two functions f; g : X -> Y are equal if f(x) = g(x) for all x in X. The set X is the domain of f and Y is the range of f. The image of f consists of the set of elements y of the range for which there exists an element x of the domain with f(x) = y. We will sometimes write f : x -> y to mean that f(x) = y.
If you look at the above you will be like ‘a lot of words for a simple thing’ , but do we totally understand the words? To resume it, a function is ‘something’ that assigns to each x and y. So it does nothing to the elements. x remains x and y remains y. There is nothing done to them, the only thing we are doing is putting an arrow between them. Like this:
So, the image says everything we need to know, but then some complicated things appear in our daily math and we start to forget what functions really are.
But after all these complicated things, in the end we can think of a function as just an arrow, that finds a connection between the elements from X to the elements from Y. And from these a lot of interesting things appear. Most of the math is based on properly understanding functions. So let yourself a couple of minutes to properly understand and visualize what is happening in a problem, theorem, axiom or definition that has functions in it.
This is a small quote from How do we learn math? by Keith Devlin:
[…] a significant proportion of university mathematics students do not have the correct concept of a function.
Do you? Here is a simple test. (This one is far simpler than the more penetrating ones Leron used.) Consider the “doubling function” y = 2x (or, if you prefer more sophisticated notation, f(x) = 2x.) Question: When you start with a number, what does this function do to it?
If you answered, “It doubles it,” you are wrong. No, no going back now and saying “Well what I really meant was …” That original answer was wrong, and shows that, even if you “know” the correct definition, your underlying concept of a function is wrong. Functions, as defined and used all the time in mathematics, don’t do anything to anything. They are not processes. They relate things. The “doubling function” relates the number 14 to the number 7, but it doesn’t do anything to 7. Functions are not processes but objects in the mathematical realm. A student who has not fully grasped and internalized that, whose underlying concept of a function is a process, will have difficulty in calculus, where functions are very definitely treated as objects that you do things do – at least sometimes you do things to them; more often, you apply other functions to them, so there is no doing, just more relating. Note that I am not claiming, and nor is Leron, that those students do not understand the difference between the two alternative possible notions of a function, or that they do not understand the correct (by agreed definition) concept. The issue is, what is their concept of a function?
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