Introduction

A set is one of the most basic building blocks of mathematics, and this chapter formalises an idea you already use informally. The concept of a set as a rigorous mathematical object was developed by the German mathematician Georg Cantor (1845-1918). In everyday language we constantly group objects together and give the group a name — a team of players, a bouquet of flowers, a bunch of keys, a flock of birds, a family of people. Mathematics borrows this same instinct but insists on one extra condition: it must be possible to say, without any doubt, whether a given object belongs to the collection or not. Consider these five collections: (i) successful persons in your city, (ii) happy people in your town, (iii) clever students in your class, (iv) days in a week, (v) the first five natural numbers. The first three are NOT examples of sets in the mathematical sense, because 'successful', 'happy' and 'clever' are relative, subjective terms — different people would disagree about who belongs to each collection, so the membership is not well-defined. The last two collections, however, ARE sets: everyone would agree on exactly which seven days make up 'days in a week', and exactly which five numbers are 'the first five natural numbers'. This distinction — whether membership can be decided unambiguously — is the entire basis for what follows: only a collection of well-defined objects qualifies as a set.