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A relation is a set, and a function is a specific kind of relation. Therefore a function is a set also.

However, at the levels of math I've studied (undergraduate only), I get the feeling people don't talk about functions as sets in the purest sense.

With the aim of better understanding whether there's some sort of informal line between functions and sets, here's one question I have:

Let $f(x)=\sqrt{x}$. (With domain all real numbers greater than or equal to 0)

Let $g(x)=\sqrt{|x|}$. (With domain all real numbers)

Is it correct to say $f \subseteq g$ ? How about $f(x) \subseteq g(x)$?

More broadly, am I right to notice a line between functions and sets? Can we always use the language of set theory to talk about functions?

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    $\begingroup$ One way to think about a function $$f: A \mapsto B$$ is to think about it as an element of the set of functions that map from the set A to the set B $\endgroup$ Jan 19, 2017 at 0:22
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    $\begingroup$ @BolutifeOgunsola: The notation is $f:A\rightarrow B$. Your symbol links the function argument with its value, e.g. $x\mapsto \sqrt x$. $\endgroup$
    – TonyK
    Jan 19, 2017 at 0:31
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    $\begingroup$ Related: mathoverflow.net/questions/90820/… $\endgroup$
    – Nick Alger
    Jan 19, 2017 at 2:56
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    $\begingroup$ You can, but that doesn't mean you should. For example, it's also true in standard set theory that $2 \subset 3$, but this is more of a coincidence (or implementation detail, to make an analogy with programming) than a mathematical truth. $\endgroup$
    – gardenhead
    Jan 19, 2017 at 5:20
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    $\begingroup$ Consider the map $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=e^x$, and consider the map $g:\mathbb{R}\to\mathbb{R}^+$ given by $g(x)=e^x$. Then $g$ is a bijection, while $f$ is not. If we formalize a function as the set of ordered pairs, we have the problem that $f$ and $g$ become the same set! You cannot reconstruct the codomain from the graph set, yet in our daily lives we consider the codomain a property of a function. This suggests we should formalize a function $f:A\to B$ as a triple $(A,B,\Gamma )$ where $\Gamma$ is the graph (set of ordered pairs). $\endgroup$ Jan 19, 2017 at 13:11

2 Answers 2

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When you think about a function as a set, it is a set of ordered pairs. The first element of the pair belongs to the domain, the second to the range. When you refer to the function as a whole, it should be $f$ or $g$, not $f(x)$ which should be the value of the function at $x$. Seen as a set of ordered pairs, every pair that belongs to $f$ also belongs to $g$, so you can say $f \subset g$

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  • $\begingroup$ Thanks for your answer, especially for the difference between f and f(x). One question: You say "Seen as a set of ordered pairs..." That makes it sound like functions can't always be thought of that way. Is there ever a situation where it's off-limits to think of a function as a set of ordered pairs? $\endgroup$
    – Archr
    Jan 19, 2017 at 0:33
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    $\begingroup$ @user404789: It's never off limits, and always valid. But there are situations where it's not useful. For instance, if you are planning a car journey, you don't need to consider how many revolutions the wheels will make. $\endgroup$
    – TonyK
    Jan 19, 2017 at 0:36
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    $\begingroup$ Some people define a function to include information about the domain and range. Your $f$ would be a function from the non-negative reals to the reals (or to the non-negative reals) while $g$ is from all the reals to the same place. It then becomes problematic to say $f \subset g$ because the function becomes a bunch of ordered pairs plus a wrapper and we don't know what to make of the subset language as it applies to the wrapper. $\endgroup$ Jan 19, 2017 at 0:37
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    $\begingroup$ @TaylorRendon: the maps you are referring to in geometry are a small subset of the functions. They are at least continuous and probably continuously differentiable. The whole set of functions from a plane to a plane is much more general. For functions in general there is no reason to suspect that nearby points are sent anywhere nearby each other. It then doesn't make much sense to talk of where a function sends a square because bits of it are sent all over the plane. Within the group of functions you are talking about, it is the same thing. $\endgroup$ Jan 28, 2021 at 0:31
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    $\begingroup$ @TaylorRendon: I think they are a very different way of thinking about functions, not just a different way. It goes back to early algebra classes where we go to the board and "draw a function". These functions are always nice and smooth, which gives many people the impression that functions must be so. Later we learn about functions as expressions and many people think every function must have a simple expression. We see lots of complaints here about using even case statements in functions. $\endgroup$ Jan 28, 2021 at 1:31
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There are two separate things:

1) How we think about mathematical objects; and

2) How we formalize mathematical objects within set theory.

People such as Archimedes, Eudoxus, Newton, Euler, and Gauss did brilliant math long before the development of set theory. Clearly set-theoretic formalization is not a prerequisite of doing math.

The formalization of mathematical objects in the context of set theory is a relatively recent historical development, having taken place between say the late nineteenth century and much of the twentieth. During that period of time mathematicians found a need for more rigorous foundations, and set theory turned out to be highly useful for this purpose. It's not necessarily the last word on foundations; nor do we spend much time caring about the formalization most of the time.

For example if we are in calculus class and we encounter the function defined by $f(x) = x^2$, we typically think of it as a machine or black box that inputs a real number and outputs the square of the input. Nobody thinks of it as a set of ordered pairs.

On the other hand if we're learning analytic geometry for the first time, given $f(x) = x^2$ we make a table containing a few sample pairs $(x, x^2)$, then we plot those points in the $x$-$y$ plane, and we see that the dots seem to form a parabola. When we graph a function we are implicitly using the idea of a function as a set of ordered pairs!

The best way to think about this is that we have two points of view, the intuition and the formalism. We go back and forth between them as needed. Sometimes a function is a machine or a mapping or a correspondence, and sometimes it's a set of ordered pairs. Whichever point of view helps us at any given moment.

One can ask about the nature of the relationship between the intuitive idea of a function (or any other mathematical object) and the set-theoretic symbology that represents or models that object. This is a question of philosophy.

Perhaps functions and numbers and sets exist in some abstract Platonic realm, and our symbols are accurate representations of them. Or perhaps our symbols are helpful but inaccurate representations. Or perhaps there are no functions or numbers or sets at all, just strings of symbols manipulated in accordance with formal rules. In that latter point of view, math is just a game like chess. Nobody thinks the laws of the universe are written in chess, but many people think the laws of the universe are written in math. Why are the rules of math so different than the rules of chess?

Many smart people have thought deeply about these issues. But when they think about these things they are doing philosophy, not math.

The takeaway is that when we do math, we use both intuition and formalism, whichever is most helpful at any given moment. And we don't think about the philosophy, unless we are doing philosophy.

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  • $\begingroup$ I don't know enough about set theory to vote on this answer, but this was an incredibly enjoyable and fascinating read! $\endgroup$ Jan 19, 2017 at 14:56
  • $\begingroup$ @DrewBuckley Thanks much. $\endgroup$
    – user4894
    Jan 19, 2017 at 23:30

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