Advanced Mathematics

3.4The Function Relation

Purpose of Section

We introduce the concept of the function, both as a “rule” that assigns a unique value to every member of a set, and from the relation viewpoint, as a subset of a Cartesian product. We discuss again the important concepts of injections, surjections, and bijections, which were introduced earlier in our study of the cardinality of sets.

3.4.1 Introduction

No doubt the concept of a function covers familiar territory for many readers of this book.¹ Normally, in the beginning of mathematics books, a function f : A → B is defined as a rule that assigns to each value x ∈ A a unique value y ∈ B. This is the definition proposed by German mathematician Peter Lejeune Dirichlet (1805–1859) in the 1830s. When we write an algebraic formula like

where x is taken as a real number, the rule is clearly understood, it assigns to each x the value sin x. We denote the function by the letter f and the value of the function at x by f(x). This motivates the Dirichlet definition of a function.

Peter Gustav Lejeune Dirichlet (1805–1859)

Dirichlet Definition of a Function

Let A and B denote sets. A function f from A to B, denoted f : A → B, is a rule that assigns to each element x ∈ A, a unique element f(x) in B. The set A is called the domain of ...