Introduction—the algebra of sets and the algebra of statements

In 1854 George Boole published The Laws of Thought, in which he showed that many of the basic ideas of logic could be expressed in symbolical terms and that laws similar to the ordinary rules of algebra could be applied to these symbols. This book was one of the most important and far reaching in the history of mathematics: for the first time it was seen that algebra, hitherto applied almost exclusively to numbers or similar mathematical entities, was in fact of much wider application. Abstract algebra could be said to have begun with Boole.

The algebra that Boole used to express logical ideas, when formalised into an abstract system in the modern manner, is now known as Boolean algebra. Besides having local applications, it is the natural algebra for work in the theory of sets, and other applications have also been discovered.

In this chapter we define in §11.2 a Boolean algebra as an abstract system and consider some simple consequences of the laws in the following section. In § 11.4 we investigate the idea of a Boolean function, continuing this study in §11.5. In §11.6 we consider some applications to logic, and in §11.7 indicate that any finite Boolean algebra may be represented as an algebra of sets. In the final sections we investigate briefly some other applications and a related idea, that of a lattice.