# The two paradigms of seeing a functions

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?

• I don't understand the distinction you are trying to draw here. The version of "function" which is taught to high school students and lower division calculus students is the "mapping" concept: a function is a machine which takes an input from one set, and produces an output from another set. A more mature notion of a function from $X$ to $Y$ is that it is a subset $f$ of the Cartesian product $X\times Y$ which has the property that if $(x,y_1)\in f$ and $(x,y_2)\in f$, then $y_1 = y_2$. Both of your notions seem to be my first. Commented Aug 25, 2022 at 22:26
• They are literally equivalent, but let's say you want to phrase something such as continuity, then we need to look what happens on the subset level not on the individual point. Or differentiability or limit or anything significant in analysis for that matter @XanderHenderson Commented Aug 25, 2022 at 22:28

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 $$f^\textrm{Inv}$$.

• Commented Aug 26, 2022 at 10:37
• I've also seen $f^{\textrm{Inv}}$ written $f^{\textrm{Pre}}$ for the preimage of $f$.
– J W
Commented Aug 26, 2022 at 11:57
• Indeed, I've also seen it happen quite often that the distinction between inverse functions and pre-images confused many first year students in mathematics. Maybe I'll give it a try next time and follow your suggestion to use different notations for them (at least at the beginning of the course). Commented Aug 27, 2022 at 20:42
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.

1. 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 $$g(\emptyset)=\emptyset$$.
• Valid points but in practical utility isn't the second view point I mentioned the one which is used 90% of the time in analysis /topology textbooks? Maybe due to my lack of knowledge I don't know of the utility of the other view points yet. Commented Aug 25, 2022 at 22:41
• Practical utility? This is abstract mathematics. The most important thing you learn from the studying this stuff is how to think better, and an important thinking skill is automatically transforming between equivalent ways of giving the same data. Do people lift weights to get better at lifting weights? No - people lift weights to get stronger or more fit. Commented Aug 25, 2022 at 23:12
• @Tryst with Freedom: I don't know of the utility of the other view points yet --- In 2006 I posted in the Math Forum discussion group a construction of a function $f:\mathbb R \rightarrow \mathbb R$ whose graph is dense in ${\mathbb R}^2.$ The existence of such functions is well known for things like nonlinear additive functions and functions such as described here, but I wanted a description that didn't rely on anything advanced (only relatively basic concepts and easily-proved results). I constructed the function as the union (continued) Commented Aug 26, 2022 at 18:46
• of various subsets of the plane (e.g. see "The function g, and here I'm identifying the function with its graph, consists of all the points in any of the sets L_n, along with ...") and I showed that the union satisfies the vertical line test. Although my original post at the ap-calculus group is no longer available (at least if it is, it's not freely available), I fortunately reposted it in sci.math where it is still freely available -- see this 5 October 2006 sci.math post. Commented Aug 26, 2022 at 18:46
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.

• I agree with the main points here, but the example of calculus/real analysis is a bit misguided - in many countries/universities it is standard to just teach rigorous real analysis (avoiding "calculus") and it still works. Calling it a "confusion" sounds like you consider the education system in your country the only possible system, which is quite disrespectful. Commented Aug 26, 2022 at 12:46