3.4.1: Formalizing the notion of functions

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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.