Links forward

Functions between sets

In the Content sections of this module, we have dealt exclusively with real functions. In the 20th century, the concept of a function was generalised to functions from any set to any other set. This has turned out to be a very powerful idea.

Definition

Let \(A\) and \(B\) be any two sets. A function \(f\) from \(A\) to \(B\) is a rule which maps each element \(x\) in \(A\) to an element \(f(x)\) in \(B\). A common notation is \begin{align*} f &\colon A\to B \\ f &\colon x\mapsto f(x) \end{align*}

Notes.

A function from \(A\) to \(B\) is often called a map from \(A\) to \(B\).
The set \(A\) is called the domain of the function, and the set \(B\) is called the codomain.
The range of a function \(f\colon A \to B\) is the set \(\{\, y\in B : y=f(x) \text{ for some } x\in A \,\}\). Another notation for the range is \(f(A) = \{\, f(x) : x\in A \,\}\).
There is a difficulty with domains. For example, consider the real function \[ f(x) = \dfrac{1}{x-2}. \] One approach is define \(A = \{\, x\in\mathbb{R} : x\neq 2\,\}\), and then define \[ f\colon A\to \mathbb{R}, \quad f(x)=\dfrac{1}{x-2}. \] A difficulty with this approach is that the precise domain must be known for the function to be properly specified.

In the next example, we use \(\mathbb{N}\) to denote the set of all positive integers (also called the set of natural numbers). That is, \(\mathbb{N} = \{1, 2, 3, \dots \}\).

Example

An infinite sequence \(a_1, a_2, a_3, a_4, \dots\) of real numbers can be thought of as a function \(S\colon \mathbb{N} \to \mathbb{R}\) given by \(S(n) = a_n\). For example:

An arithmetic sequence is of the form \[ a,\ a+d,\ a+2d,\ a+3d,\ \dots \] for some \(a,d \in \mathbb{R}\). The associated function \(S\colon \mathbb{N} \to \mathbb{R}\) is given by \(S(n) = a+(n-1)d\).
A geometric sequence is of the form \[ a,\ ar,\ ar^2,\ ar^3,\ \dots \] for some \(a,r \in \mathbb{R}\). The associated function \(S\colon \mathbb{N} \to \mathbb{R}\) is given by \(S(n) = ar^{n-1}\).

Exercise 6

Classify each of the following sequences as arithmetic, geometric or neither.

\(2,\ 5,\ 8,\ 11,\ 14,\ \dots\)
\(3,\ -6,\ 12,\ -24,\ 48,\ \dots\)
\(1,\ 0.1,\ 0.01,\ 0.001,\ 0.0001,\ \dots\)
\(1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ \dots\)
The sequence of decimal approximations to \(\pi\): \[ 3,\ 3.1,\ 3.14,\ 3.142,\ 3.1416,\ 3.14159,\ \dots\,. \]

Example

Consider the function \(f\colon \mathbb{R}\to \mathbb{R}\) given by \(f(x)=\lfloor x \rfloor\). The range of \(f\) is \(\mathbb{Z}\). Hence \(f\) is also a function from \(\mathbb{R}\) to \(\mathbb{Z}\).

Exercise 7

Determine the domain, codomain and range of \(f\colon \mathbb{N} \to \mathbb{N}\) given by \(f(n) = 2n+1\).

Next page - Links forward - Functions between finite sets