Boolean algebra

Abstract

In his book The Laws of Thought (1854) George Boole discovered that algebraic formulas and arithmetic operations can be interpreted so that they cover ordinary logic. One of his aims was to analyze the complicated statements and long lines of reasoning of philosophers. In this he was not successful since, as he says himself, their basic concepts are too vague to lend themselves to mathematical treatment. Under more precise circumstances as in the analysis of complicated combinations of simple statements he was more successful. Today, the machinery he invented, the Boolean algebras and rings, is used to analyze switching circuits. This chapter is just a simple account of finite Boolean algebras with a last section on the equivalence of Boolean functions under permutation and complementation. It does not touch the important questions of the economy of construction and complexity of circuits.