In mathematics a set is a collection of elements, especially a collection having some feature or features in common. The set may have a finite number of elements, e.g., the set of prime numbers less than 100; or it may have an infinite number of elements, e.g., the set of right triangles. The elements of a set may be anything at all--from simple integers to arbitrarily complex objects. However, all the elements of a set are distinct--a set may contain only one instance of a given element.
For example, , , , and are all sets the elements of which are drawn from . The set of all possible elements, U, is called the universal set . Note also that the elements comprising a given set are not ordered. Thus, and are the same set.
There are many possible operations on sets. In this chapter we consider the most common operations for combining sets--union, intersection, difference:
Figure illustrates the basic set operations using a Venn diagram . A Venn diagram represents the membership of sets by regions of the plane. In Figure the two sets S and T divide the plane into the four regions labeled I-IV. The following table illustrates the basic set operations by enumerating the regions that comprise each set.
Figure: Venn diagram illustrating the basic set operations.