Relations and Functions

Abstract

Functions are important in many areas of mathematics. Elementary algebra starts to differ from arithmetic when the concept of a function is developed. Calculus can be viewed as being the study of functions, with special emphasis on the continuous properties and of certain ways of associating new functions with a given one. Binary relations can be seen as a generalization of functions.

In the first section, binary relations are defined as subsets of the direct product of two sets. Operations on relations (union, intersection, composition) are studied, as are the properties of reflexivity, symmetry, and transitivity. Equivalence and order relations are defined and studied. The study of functions is introduced by looking at special relations. Operations such as composition are discussed. Finally, one-to-one functions, onto functions, and inverses are considered.