Throughout mathematics there are countless instances where the concepts of "element" and "set of elements" (or class)  play a crucial role. Every freshman student in mathematics is familiar with the set of integers, the set of all right triangles, the set of lines perpendicular to a given plane, and the set of points on a line. The concept of set is not limited to mathematics, however. The totality of books in a library, of people in a room, and of fish in a given stream are examples of sets. The purpose of this chapter is to investigate the nature of sets and the ways in which they may be combined. That sets obey laws of algebra similar, although not identical, to the laws of algebra for real numbers may seem strange at first. However, it will be shown how this phenomenon is a natural and very useful one.

In any subject in mathematics there are certain terms so basic that definition is impossible. In plane geometry, the terms point and line are undefined, although a student of geometry is encouraged to form an intuitive notion of the meaning of these words. We will take as undefined terms for the algebra of sets the words element and set. Intuitively we think

of elements as the basic objects which, in collections, constitute sets. As symbols we shall use the letters of the alphabet in lower case italics (a, b,  c, x, y, etc.) to represent elements, and capital letters (A, B, X, etc.) to represent sets. A further symbol, ∈, will be used to denote an undefined relation which may or may not hold between a particular element and a particular set, in that order. We may write, for example, m ∈ X, and read this symbol "m is a member of the set X." It will be assumed that for each element m and each set X in any discussion it is possible to determine whether or not the relation m ∈ X is valid.

We will say that set X equals set Y, and write X = Y, if and only if the two sets are identical, that is, contain exactly the same elements. If a set X consists entirely of elements which are members of a second set Y, we say that X is a subset of Y and write X ⊆ Y. If, in addition, Y contains one or more elements not in X, we say X is a proper subset of Y.

It is convenient to introduce names for two special sets which will be important in any application. The first is called the universal set and is defined to be the set consisting of all elements under discussion. This set is also referred to as the domain of discourse, or the fundamental domain.

The universal set will be denoted by the symbol 1. We note that every set is a subset of the universal set. The second special set, called the null set, is defined to be a set containing no elements at all. By definition, the null set is a subset of every other set. The notation for the null set will be the symbol 0. It is important to note that 0 and 1, as used here, are not numbers but the names of two special sets.

The algebra that will be developed is an algebra for sets, not for elements of sets. For example, the symbol m ∈ X cannot be introduced into the algebra. It is frequently important to work with individual elements of sets, and since we cannot handle elements as such in the algebra, it is convenient to introduce the concept of a unit set. A unit set is a set which consists of a single element, and if this element is, say, x, we denote the set by {x}. In other instances as well, if the set is specified by listing each of the elements in the set, the symbol { } will be used. For example,  [a, b, c) is understood to be the set consisting of the elements a, b, and c only.

Associated with each set X is another set X' called the complement of X and defined to be the set consisting of all elements of the universal set which are not elements of X. As special cases we note that the null set and he universal set are each complements of the other.

Example. Consider a stack of books of which some are bound in red, some in black, and the rest in yellow. Suppose that all red books and some of the black

books are written in English. The remainder of the black books are in German,  and the yellow books are written in French. Let the set of all books in the stack

be the universal set and let other sets be denoted as follows:

R is the set of red books,

Y is the set of yellow books,

B is the set of black books,

E is the set of books written in English,

F is the set of books written in French,

G is the set of books written in German.