Set Theory

A set is a collection. As with any collection, a set is composed of objects, called members or elements. The elements of a set may be physical objects or mathematical objects. A set may be composed of baseball cards, salt shakers, tropical fish, numbers, geometric shapes, or abstract mathematical constructs such as functions. Even ideas may be elements of a set. In fact, the elements of a set are not required to have anything in common except that they belong to the same set. The collection of all the junk at a rummage sale is a perfectly good set, but one in which few of the elements have anything in common, except that someone has gathered them up and put them in a rummage sale.

In order to specify a set and its elements as completely and unambiguously as possible, standard forms of notation (sometimes called set-builder notation) have been adopted by mathematicians. For brevity a set is usually named using an uppercase Roman letter, such a S. When defining the set S, curly brackets are used to enclose the contents, and the elements are specified, inside the brackets. When convenient, the elements are listed individually. For instance, suppose there are five items at a rummage sale. Then the set of items at the rummage sale might be specified by R =. If the list of elements is long, the set may be specified by defining the condition that an object must satisfy in order to be considered an element of the set. For example, if the rummage sale has hundreds of items, then the set R may be specified by R =. This is the set of all x such that x is a real number, and 0 is less than x, and x is less than 1. The special symbol ø is given to the set with no elements, called the empty set or null set. Finally, it means that x is an element of the set A, and means that x is not an element of the set A.