Boolean Algebra

A set is a collection of objects, called members or elements. The members of a set can be physical objects, such as people, stars, or red roses, or they can be abstract objects, such as ideas, numbers, or even other sets. A set is referred to as the universal set (usually called I) if it contains all the elements under consideration. A set, S, not equal to I, is called a proper subset of I, if every element of S is contained in I. This is written and read "S is contained in I." (see Figure 1)

If S equals I, then S is called an improper subset of I, that is, I is an improper subset of itself (note that two sets are equal if and only if they both contain exactly the same elements). The special symbol is given to the set with no elements, called the empty set or null set. The null set is a subset of every set.

When dealing with sets there are three important operations. Two of these operations are binary (that is, they involve combining sets two at a time), and the third involves only one set at a time. The two binary operations Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group. are union and intersection. The third operation is complementation. The union of two sets S and T is the collection of those members that belong to either S or T or both. (see Figure 2)

The intersection of the sets S and T is the collection of those members that belong to both S and T. (see Figure 3)

The complement of a subset, S, is that part of I not contained in S, and is written S'. (see Figure 4)