# 3.4.1: Formalizing the notion of functions

In mathematics, of course, we need to work with functional relationships in the abstract. To do this, we introduce the idea of function. You should think of a function as a mathematical object that expresses a functional relationship between two sets. The notation f : A B expresses the fact that f is a function from the set A to the set B. That is, fis a name for a mathematical object that expresses a functional relationship from the set A to the set B. The notation f : A B is read as “ f is a function from A to B” or more simply as “ f maps A to B”.

Mathematical functions are different to functions in a programming language in Java. We’ll come back to this in the next section.

If f : A B and if a A, the fact that f is a functional relationship from A to B means that f associates some element of B to a. That element is denoted f (a). That is, for each a A, f (a) ∈ B and f (a) is the single, definite answer to the question “What element of B is associated to a by the function f ?” The fact that f is a function from A to B means that this question has a single, well-defined answer. Given a A, f (a) is called the value of the function f at a.

For example, if I is the set of items for sale in a given store and M is the set of possible prices, then there is function c : I M which is defined by the fact that for each x I, c(x) is the price of the item x. Similarly, if P is the set of people, then there is a function m: P P such that for each person p, m(p)is the mother of p. And if $$\mathbb{N}$$ is the set of natural numbers, then the formula

$$s(n)=n^{2}$$ specifies a function s: $$\mathbb{N} \rightarrow \mathbb{N}$$.

It is in the form of formulas such as $$s(n)=n^{2}$$ or $$f(x)=x^{3}-3 x+7$$ that most people first encounter functions. But you should note that a formula by itself is not a function, although it might well specify a function between two given sets of numbers. Functions are much more general than formulas, and they apply to all kinds of sets, not just to sets of numbers.