Advantages of considering a function as a set

Question

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.

Answer 1

You can see :

• John von Neumann, Eine Axiomatisierung der Mengenlehre, (1925), translated as An axiomatization of set theory, into : Jean van Heijenoort (editor), From Frege to Godel : A Source Book in Mathematical Logic (1967), page 393-on :

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 $[f,x]$.

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You can see also :

• Hendrik Pieter Barendregt, The Lambda Calculus : Its Syntax and Semantics (rev ed - 1984), page 3 :

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.

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For Category theory, you can see :

• Steve Awodey, Category Theory (2nd ed : 2010), page 3 :

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.

Answer 2

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.