Algebra of Classes. Algebra of Logic

Abstract

Consider a set I which, in the text which follows, will play the role of a universal set. The different subsets of I form a set on which may be defined three operations: union, intersection and complementation. Endowed with these, the set of subsets of I then constitutes an ‘algebra’, usually known as an “algebra of sets” or an “algebra of classes”. From the definitions of set, subset and those of the three operations above the diverse properties of this algebra can be demonstrated. The more important of these properties will be stated later. They will be justified with the aid of Venn diagrams. However, by way of an exercise the reader can carry out rigorous abstract demonstrations based on the definitions in Chapter 1. It will be shown by an example (relationship E₉ below) which type of demonstration can then be made.