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Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to define a function without being acquainted with any concept of Cartesian Products.

To answer this question I told him the definition of a function may be given by defining a function $f$ from the set $X$ to $Y$, denoted by $f:X\to Y$, to be some "rule" (the word rule being an undefined concept) which associates, to each member of $X$, exactly one element of $Y$.

But my friend said that the definition using the concept of "rule" isn't rigorous and there are examples that don't obey the definition but still is a function in the Cartesian Product sense.

My questions are,

  1. Why the definition of function using concepts such as "rule" isn't rigorous?

  2. Is there really any example which doesn't obey the definition but still is a function in the Cartesian Product sense?

Git Gud
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    It is not rigorous unless you give a precise definition of "rule". But without such a definition, it is also impossible to find examples violating it. – Tobias Kildetoft Dec 17 '14 at 12:56
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    Choice functions can be examples for 2. – Git Gud Dec 17 '14 at 12:57
  • As for 1. the definition of function you propose was used for a long, long time, until mathematicians found it necessary to give a more precise meaning to the concept function, (I assume there are concepts nowadays which are used with no precise definition and which mathematicians will eventually find necessary to formalize a little better). With regards to current mathematical practice, I agree with your friend because rule isn't a primitive concept in $\sf ZFC$ and so you need to define it in $\sf ZFC$. – Git Gud Dec 17 '14 at 12:59
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    You can see The Lambda Calculus for a mathematical theory based on the primitive concept of function as "rule". But this theory "assume" it as primitive and thus does not provide an "explicit" definition of function (i.e. through a reduction of it to already defined concepts). – Mauro ALLEGRANZA Dec 17 '14 at 13:00

2 Answers2

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There is an informal concept of "function", which says that a function is any correspondence (itself an informal term) that assigns one definite output value to each input value of the function.

This informal concept has various formalizations, depending on the precise use one has for it.

  • The set-based concept of a function as a set of ordered pairs such that there is nothing that is the first element of two different pairs in the set. Such a function has a set as its domain and a set of its range, and it is a subset of the Cartesian product of the domain and the range.

    Here, in particular, it doesn't matter where we get the set from, as long as it satisfies the defining condition. In particular, the set-theoretic Axiom of Choice states that there exist function with particular properties without giving any kind of "rule" for how to find the output for a particular input.

The set-based formalization is be convenient and expressive enough that it is the default choice for formalizing functions in many areas of mathematics, together with the general tendency to formalize everything in set theory. But it is not the only one possible, and in some situations a different formalization is more useful:

  • In logic and related areas is is common to model a function simply as a symbol together with axioms and equations that state properties that we want the symbol to have. For example, the Peano Arithmetic axioms for the natural numbers posit that there is a successor function and addition/multiplication functions that are not considered objects in themselves but simply symbols that can be manipulated according to certain rules. Intuitively, these symbols are to be thought of as realizing the informal concept of a function.

    This is not as expressive as the usual set-based concept, but is convenient because it can be used in settings where one wishes to avoid depending on set theory for one reason or another.

    In higher-order logic, one can even quantify over this kind of functions.

  • Also in logic, sometimes a "function" is considered to be simply a logical formula with two variables, $\varphi(x,y)$ such that $\forall x,y,z:\varphi(x,y)\land \varphi(x,z)\to y=z$ is true (or provable, depending on your needs).

    In some ways, this is stronger than the usual formalization, which is necessary for some purposes within set theory itself -- for example, the successor function for ordinal numbers can be expressed as a formula but is not a set in standard set theory. On the other hand, if we allow additional parameters in $\varphi$ we can express every set function, because $(x,y)\in f$ is a good formula when $f$ is a set that represents a function in the ordinary way.

    On the other other hand, viewing functions as formulas means that one cannot quantify over functions (that is, one cannot say that such-and-such holds for all functions of a certain kind, nor that there is some function with such-and-such property).

  • In lambda calculus, used in computer science and related areas, the basic kind of object is a function, represented as an actual expression for how to compute function values, wrapped up as a mathematical object in its own right.

    This view is useful for reasoning about computable functions rather than the unlimited "wild" functions that the set-theoretic view lets us speak about. Being less expressive allows this formalization to allow more convenient reasoning of certain kinds.

The problem with saying that a function is just a "rule" is that the term "rule" itself needs a definite meaning before that concept says any more than the informal "correspondence" definition. For example, if we say

$f(n)$ is the length in words of the shortest English phrase that describes $n$ unambigously.

is that then a "rule"? If it is, then fairly soon we're hit by Berry's paradox. But if it is not, then the question of what exactly a valid "rule" is, demands to be answered. One may propose various answers, of course, but by their nature such an answer will lead to a concept of function that is something more than just saying "a rule".

hmakholm left over Monica
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  • for example, the successor function for ordinal numbers can be expressed as a formula but is not a set in standard set theory. This is less a problem with the set-theoretic definition of a function, and more a problem with ZFC specifically. A set theory such as NBG does not have this problem, because a function is simply a proper class. Also, the second definition you explained is actually a special case of the first. – Angel Oct 20 '21 at 19:15
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You can indeed formally define a function in terms of Cartesian products:

A subset $f$ of the Cartesian product $A\times B$ is said to be function mapping the elements of $A$ to to the elements $B$ if and only if for all $x\in A$ there exists a unique $y\in B$ such that $(x,y)\in f$.

You might also formally define functions as a certain kind of relation, but then you would you would probably have to define relations in terms of Cartesian products or some equivalent, e.g. arrow diagrams or graphs.

If you simply want to formally state that $f$ is a function mapping the elements of set $A$ to the elements of set $B$, I have found it very useful to write:

For all $x\in A$, we have $f(x)\in B$.

So, to answer your questions:

  1. Why the definition of function using concepts such as "rule" isn't rigorous?

Edit: Unlike the notions of Cartesian products and subsets, the notion of a "rule" is not formally defined in set theory and logic. There are no axioms about "rules". The closest you come is the selection criterion for a subset, but you only need to specify one if you are talking about a particular subset.

  1. Is there really any example which doesn't obey the definition but still is a function in the Cartesian Product sense?

No, but perhaps your friend is thinking of the notion of a partial function that is similar.

9 years later...

From Terence Tao's, "Analysis I," p. 49, we have the definition:

Definition 3.3.1 (Functions). Let $X$, $Y$ be sets, and let $P(x,y)$ be a property pertaining to an object $x \in X$ and an object $y \in Y$, such that for every $x \in X$, there is exactly one $y \in Y$ for which $P(x,y)$ is true (this is sometimes known as the vertical line test). Then we define the function $f : X \to Y$ defined by $P$ on the domain $X$ and range $Y$ to be the object which, given any input $x \in X$, assigns a output $f(x) \in Y$, defined to be the unique object $f(x)$ for which $P(x, f(x))$ is true. Thus, for any $x \in X$ and $y \in Y$, $$y = f(x) \iff P(x,y)\,\text{is true.}$$

Here, the "rule" is given by the binary predicate $P$

Dan Christensen
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