The two paradigms of seeing a functions

Question

When we are first taught functions , we are typically taught of them as maps between real numbers and we taught to think of them mainly as a mapping between elements. It seems intuitive to take this route as well because most of the functions we are familiar with such as the square root, square etc are defined as a general which can taken any element in the real number. Eg: $f(x)= \sqrt{x}$ , $f(x) = x^2$.

However, to lead to the ideas in analysis and topology, one must think of function rather as a mapping between subsets. Now, the initial idea of course is equivalent to this, but it is usually non obvious why this form is more useful. Yet, I have never seen this point being emphasized when functions are being taught early on.

Why is it so that this second way is not emphasized? Has there been any attempts to change into this alternate pedagogy for functions?

Answer 1

The functions $f: A \to B$, $f^{-1}: B \to A$, $f^{\textrm{Im}}:2^A \to 2^B$, and $f^{\textrm{Inv}}: 2^B \to 2^A$ are all related, but distinct, functions.

We often abuse notation and use the same notation for $f$ and $f^{\textrm{Im}}$, and for $f^{-1}$ and $f^{\textrm{Inv}}$. This is just for convenience though: not because we actually think these are the same functions. It is done to save time writing notation, and because it is entirely clear from context which function is actually meant.

For instance if $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = x^2$, then

• $f(2) = 4$

• $f^{\textrm{Im}}([1,2]) = [1,4]$

• $f^{-1}$ is not a function and

• $f^{\textrm{Inv}}([1,4]) = [-2,-1] \cup [1,2]$

You might see the following technically incorrect notations in a book:

• $f^{-1}(4) = \{-2,2\}$ (they should really write $f^{\textrm{Inv}}(\{4\}) = \{-2,2\}$)
• $f^{-1}([1,4]) = [-2,-1] \cup [1,2]$ (they should really be using $f^{\textrm{Inv}}$)
• $f([1,2]) = [1,4]$ (they should really by using $f^{\textrm{Im}}$).

We abuse notation like this to save time, finger strength, and mental energy. It is unlikely to lead to genuine logical errors because what is actually meant is clear from the type of the input. This is similar to type coercion in programming.

p.s. My favorite notation is actually $\vec{f}$ for $f^{\textrm{Im}}$ and $\overleftarrow{f}$ for $f^\textrm{Inv}$.

Answer 2

1. Because there isn't just a second way, but also a third way, and a fourth way, and many others.

For example, we can also think of a function $f: X \rightarrow Y$ as a function that takes an element of $Y$ and gives all the elements of $X$ that map to it - i.e. the data of $f$ is equivalent to the data of $f^{-1}$, where $f^{-1}$ is thought of as a function from $Y$ to the set of subsets of $X$.

Another way is to think of a function $f: X \rightarrow Y$ as a function from subsets of $Y$ to subsets of $X$.

I can keep coming up with more complicated examples if you would like.

The point is that, eventually, the student has to be able to see many different ways of representing the same data as being equivalent, and pointing out one specific way isn't all that helpful.

2. If we have a function $g: 2^X \rightarrow 2^Y$, it does not necessary come from a function $f: X \rightarrow Y$. For example, if $g$ comes from such a function $f$, we must have $g(\emptyset)=\emptyset$.

Answer 3

1. I don't understand the difference you are claiming.

2. Why does it surprise you that an advanced concept is NOT used at the intro? It is a normal part of pedagogy to teach harder things later. Your question springs from the same confusion as people who ask why not teach real analysis before calculus. The literal reason is because humans have organic brains and don't learn everything immediately, by definitions. Throw in also that the vast majority will not be math majors and won't need analysis or topology, ever. Something that seems crushingly obvious, but is routinely ignored by the know mathematics, don't know audience, crowd.