Introduction

Discussion 1.1.1 We shall begin by giving an informal description of some of the topics which appear in Chapter 1. The central concept is that of an ordered set. Roughly, an ordered set is a collection of items some of which are deemed to be greater or smaller than others. We can think of the set of natural numbers as an ordered set, where, for example, 5 is greater than 2, 0 is less than 100, 1234 is less than 12687 and so on. We shall see later that one way in which the concept of order arises in computer science is by regarding items of data as ordered according to how much information a certain data item gives us. Very crudely, suppose that we have two programs P and P′ which perform identical tasks, but that program P is defined (halts with success) on a greater number of inputs than does P′. Then we could record this observation by saying that P is greater than P′. These ideas will be made clearer in Discussion 1.5.1. We can perform certain operations on ordered sets, for example we have simple operations such as maxima and minima (the maximum of 5 and 2 in the ordered set of natural numbers is 5), as well as more complicated ones such as taking suprema and infima. If the reader has not met the idea of suprema and infima, then he will find the definitions in Discussion 1.2.7.