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Can anyone give me some reference (could be a book, paper, or even notes), in which the author writes about the advantages of defining a function as a set for mathematics in general?

Let $f(x) = x + 1$ be the law of a function.

Let $X \times Y \neq \varnothing$

(a) We can write $f = \{(x; x+1) : x \in X\}$

(b) The fact is that, without this language we can understand a function as a simple law that associates a number to another, or even in more informal ways as a machine that transforms nunbers.

I need a material that shows why (a) it's better than (b) for mathematics in general.

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    $\begingroup$ What is $Y$ in your question? $\endgroup$ Dec 5, 2015 at 17:31
  • $\begingroup$ I just wanted to show that the function emerges of the Cartesian product of this two sets $X$ and $Y$. or $f \subset X \times Y$. $\endgroup$ Dec 5, 2015 at 17:37
  • $\begingroup$ Economy of thought. We can start from function as primitive; see Category Theory. $\endgroup$ Dec 5, 2015 at 17:39
  • $\begingroup$ When you write the function as a set of tuples or vectors this is named, in some books, the graph of the function $f$, not the function itself. $\endgroup$
    – Masacroso
    Dec 5, 2015 at 18:21
  • $\begingroup$ So you are saying that in this case $Y=\{x+1\mid x\in X\}$? Or $Y=\Bbb R$ or $\Bbb C$? In some definitions of function, that matters. That shows why a formal, clear definition is necessary. $\endgroup$ Dec 5, 2015 at 18:25

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You can see :

We prefer, however, to axiomatize not "set" but "function". The latter notion certainly includes the former. [...] We consider two domains of objects, that of " arguments" and that of" functions". (Both words are, of course, to be taken in a purely formal way, as if they had no meaning.) The two domains are not identical, but they partly overlap. (There are "argument-functions", which belong to both domains.)

Now a two-variable operation $[x,y]$ (read "the value of the function $x$ for the argument $y$"), whose first variable $x$ must always be a "function" and whose second variable $y$ must always be an "argument", is defined in these domains. What is formed by means of it is always an "argument", $[x,y]$.

The operation $[x,y]$ corresponds to a procedure that is that is encountered everywhere in mathematics, namely, the formation, from a function $f$ (which must be carefully distinguished from its values $f(x)$!) and an argument $x$, of the value $f(x)$ of the function $f$ for the argument $x$. Instead of $f(x)$ we write $[f,x]$.


You can see also :

The lambda calculus is a type free theory about functions as rules, rather than as graphs. "Functions as rules" is the old fashioned notion of function and refers to the process of going from argument to value, a process coded by a definition. The idea, usually attributed to Dirichlet, that functions could also be considered as graphs, that is, as sets of pairs of argument and value, was an important mathematical contribution. Nevertheless the $\lambda$-calculus regards functions again as rules in order to stress their computational aspects.


For Category theory, you can see :

What is category theory? As a first approximation, one could say that category theory is the mathematical study of (abstract) algebras of functions.

We begin by considering functions between sets. I am not going to say here what a function is, anymore than what a set is. Instead, we will assume a working knowledge of these terms. They can in fact be defined using category theory, but that is not our purpose here.

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This is so deeply ingrained since the early 20th century that it is hardly ever argued explicitly anymore. For one thing, it does away with the logical absurdities that arise from such horrible concepts as "multivalued functions". It is very hard to keep a consistent notation if you start from a general relationship (such as an algebraic equation in $x$ and $y$) and try to prove that it actually defines a function $y=f(x)$ unless you somehow identify functions with their graphs.

Please note that the usual definition of a relation (function or otherwise) is not that is is a graph: to be consistent the definition is an ordered triplet consisting of two sets and a subset of their cartesian product.

(Update based on some comments)

In secondary school teaching the recent trend is to avoid sets as an explicit concept. In that context useful metaphors for single-valued functions are black boxes with an input and output channel, and even little hand cranks illustrating the "work" of a tranformation.

For a balanced view on the interplay between these two approaches, see for instance section 12.5 in B. Baumslag, "Fundamentals of Teaching Mathematics at University Level," Imperial College Press 2000.

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  • $\begingroup$ "This is so deeply ingrained .. that it is hardly ever argued explicitly" Perhaps that's a problem and it should be stated more explicitely. I first encountered functions in the second grade as "function boxes" where you plug something in and and apply a rule (the drawings even had little handcranks on them). When I first encountered them as sets in graduate school it threw me for a loop. Especically in topology as transformations. Would have really helped if I had thought of them as equivalently as sets as well. $\endgroup$
    – fleablood
    Dec 6, 2015 at 0:39
  • $\begingroup$ Regarding the "usual definition of a relation or function", you can see also What is the right way to define a function?. $\endgroup$ Dec 6, 2015 at 8:04
  • $\begingroup$ @fleablood Teaching functions without set theory traces its pedigree to the venerable Dirichlet; but set theory really simplifies the formalism when turning the handcrank only once can result in several little packages coming out of the box. $\endgroup$ Dec 6, 2015 at 11:56
  • $\begingroup$ @fleablood: Your comment shows how different things have apparently become. I don't think I heard the mathematical term "function" (at school, at least; this is in the U.S.A.) until 7th grade, maybe not until 8th grade (ages 13-14; around 1971-73), but functions were defined as sets of ordered pairs pretty much from the beginning. Certainly they were done this way in my 9th grade Algebra 1 class (used Dolciani's Modern Algebra Structure. Book One), and functions were even treated this way in the college algebra and precalculus texts I taught from throughout the 1980s. $\endgroup$ Dec 7, 2015 at 15:12
  • $\begingroup$ "apparently become"? New Math did not last that long and was not deemed successful and has become a forgotten footnote in history since. $\endgroup$
    – fleablood
    Dec 7, 2015 at 16:33

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