Functions

Abstract

This chapter reviews the fundamental properties of functions. After this concept has been defined, basic concepts such as the domain, codomain, range, image, preimage and arity of a function are introduced. Through the notion of graphs, functions are then related to sets of ordered pairs. Next, injective (one-to-one), surjective (onto) and bijective functions are defined and various of their basic properties are derived, such as the fact that bijective functions have inverses. Function composition is introduced and its algebraic properties are investigated. The connection between bijections and cardinalities of sets is explored, including the Schröder-Bernstein theorem and Cantor’s diagonalisation argument. Fixed points of monotonic functions are also explored, with a presentation of the Knaster-Tarski theorem and a method for computing fixed points by iteration.