Data Structures and Algorithms
with Object-Oriented Design Patterns in C# |

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:

**union**-
The
*union*(or*conjunction*) of sets*S*and*T*, written , is the set comprised of all the elements of*S*together with all the elements of*T*. Since a set cannot contain duplicates, if the same item is an element of both*S*and*T*, only one instance of that item appears in . If and , then . **intersection**-
The
*intersection*(or*disjunction*) of sets*S*and*T*is written . The elements of are those items which are elements of*both**S*and*T*. If and , then . **difference**-
The
*difference*(or*subtraction*) of sets*S*and*T*, written*S*-*T*, contains those elements of*S*which are*not also*elements of*T*. That is, the result*S*-*T*is obtained by taking the set*S*and removing from it those elements which are also found in*T*. If and , then .

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.

Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.