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Since a few days ago, I have been working with Schaum's Outline of General Topology by Seymour Lipschutz. So far I have been studying the first chapters about sets and functions to review, and to make sure that I know his notation.

My question is more philosophical in nature, and would perhaps strike you as "silly" but I will pose it anyway, and would love to hear your thoughts!

In chapter 2, he defines a function in this way (rather wordily):

Suppose that to each element of a set $A$ there is assigned a unique element of a set $B$; the collection, $f$, of such assignments is called a function from $A$ into $B$ ... (p.17, emphasis added)

Very standard I guess, but what caught my attention was the word assign. What does it really mean to "assign" something to something else? Where (i.e., in what kind of set) is this assignment stored?

My intuition about assignment was (and still is) that it is simply a pair of elements; i.e., an assignment of elements $a \in A$ to $b \in B$ is simply a subset of $A \times B$. However, the (philosophical) problem comes from Lipschutz's next statement:

To each function $f: A \rightarrow B$ there corresponds the relation in $A \times B$ given by $$\{ ( a,f(a) ) \vert a \in A\}.$$ (p.17, emphasis added)

Thus to the function corresponds a relation, i.e., the function and the relation are seen as different objects. The problem is then metaphorically swept under the carpet by not "distinguish[ing] between a function and its graph." I interpret this a bit colorfully as "they are different, but we are not supposed to ask questions about it."

I remember when I studied my first algebra course, then a function $f: A \rightarrow B$ was indeed defined as a special case of a relation, i.e., as a subset of $A \times B$. I never thought much about it then, but now I see that doing it this way avoids any reference to some "assignment" and we know exactly which set the function "lives in" -- $A \times B$, so we don't have to think about where the "assignment" is "stored." But by making a difference between the function and the relation (although there is a map between them, which is my interpretation of "corresponds") the (philosophical) question of the nature of this "assignment" arises (at least in my mind).

I guess another way to phrase what I'm thinking about is that in my mind, "assignment" is done by means of a map, or function, between sets; but what does it mean then to define functions in terms of some "assignment" operation?

I apologize in advance for taking up your time with this! (I feel so stupid for thinking about these kinds of things instead of actually working with the topology problems...). But I'm wondering if there is some definition or notion of what "assignment" means in this context? Or perhaps it is just some language that we shouldn't think more about? Or perhaps there is something I have missed, not being a native English speaker?

If you have any insights I would love to hear them :)

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    $\begingroup$ It's just handwaving to avoid talking about functions as sets of ordered pairs. $\endgroup$ – Angina Seng Aug 21 '20 at 15:32
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    $\begingroup$ Note that $f\colon\{1\}\to\{1\}$ with $f(1)=1$ and $g\colon\{1\}\to\{1,2\}$ with $g(1)=1$ have the same graph $\{(1,1)\}$ but are not the same function. A functions codomain is not determined by the graph and hence the function carries more information than just a set of ordered pairs. $\endgroup$ – Christoph Aug 21 '20 at 15:44
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    $\begingroup$ Defining functions as their graphs (i.e. sets of ordered pairs) is the simplest precise way to do it, but is not necessarily the optimal way to (intuitively) think about functions in many mathematical contexts. So in introductory texts, functions may be described as "assignments" or "associations" or "machines" etc because it is easier to apply them that way. For example, the fundamental operation of function composition is easier to understand, for a newcomer, with functions thought of as assignments rather than sets of ordered pairs. $\endgroup$ – Ned Aug 21 '20 at 18:20
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    $\begingroup$ It is just the meta-mathematical way of defining a function $\endgroup$ – Mathphys meister Aug 22 '20 at 21:31
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    $\begingroup$ I think that if you were to really press the author on this point, they would stop using the word "assign" and just tell you that a function is defined to be a set of ordered pairs satisfying certain properties (or they would give some similar precise definition of a function). But that more precise definition might seem scary to less mathematically mature readers. Or the author might think it distracts from the flow of the chapter to give the fully precise definition. $\endgroup$ – littleO Aug 22 '20 at 21:31
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There are a few ways to formally define functions, and a few ways to think about assignment. But it's in a textbook author's best interest not to pick sides when it's not relevant to what they plan to do with functions.

Definitions

It's generally agreed that the "graph" of a function $f:A\to B$ is the subset of $A\times B$ that you could write as $\left\{\left(a,f(a)\right)\left|a\in A\right.\right\}$. There are a number of notations for this, but I'll use $G(f)$ for the graph of $f$.

Some texts will say that a function is its graph. This fits in well with a discussion of relations (e.g. some relations are functions and some are not). This means that you can't derive the intended codomain (the set $B$) in "$f:A\to B$" from a function, so that whether a function is onto/surjective is not an inherent property of the function, but a property of the function and whatever target/codomain was mentioned in a given context, together. In practice, that is usually fine; but if you want to talk about two functions being equal or not, you might not want to go that route.

Other texts will bundle the domain and codomain with the graph as part of the data of the function. So a function $f$ would be something like the ordered triple $\left(A,B,G(f)\right)$. This way function is either surjective or it isn't. And functions with different codomains are definitely different objects.

Almost no texts will do this, but since you can recover the domain from the graph (the exact way you would do that depends on how you set up your pairs in set theory), you could throw out the domain, and say that a function is $(G(f),B)$ or similar.

Assignment

Whether or not the domain and codomain are bundled with the graph or not, "assignment" of outputs to inputs in this sort of context usually just means the graph exists, as a set. Every first-coordinate is "assigned" to the corresponding second-coordinate.

But you might be thinking about a rule to calculate the output from the input or a logical fact that tidily describes the entire graph. If you/someone is thinking about those sorts of things, then you're not thinking about all functions, but perhaps something like "computable functions" or "constructible functions" or "definable functions".

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