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

    For example, if is the set of items for sale in a given store and is the set of possible prices, then there is function → which is defined by the fact that for each ∈ Ic(x) is the price of the item x. Similarly, if is the set of people, then there is a function m→ such that for each person pm(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.