Set Theory

Set theory refers to a system for mathematically reasoning about sets, the primary subject of set theory and taken as a primitive term. There are several axiomatic systems for describing sets and set theory. This page will not give any axiomatic system in particular, but will outline the common relationships in set theory and give the notations used in discussing sets.

Sets and Elements
Set and element will be given as primitive terms. Sets are generally thought of as collections of elements.

Element e is a member of set S:



Element e is not a member of set S:



There are two basic methods are used for describing and defining sets. The first is to explicitly list the elements of the set, and the second is to define a common property held by all elements of the set.

Definition of a set S containing elements a, b, and c:

Definition of a Set S as containing all elements of set A with the property P(x):

Common Sets of Numbers
The following Symbols are used to give the sets containing all numbers of a given classification. They are not completely standardized from text to text or in practice, and many more can be defined. The sets of the Natural and Whole Numbers are particularly problematic in this respect, as they are often confused and/or merged. For example, the the Gersting and VanDrunen books taught in this class at the time of this writing give different definitions for the Sets of the Natural and Whole Numbers, and the Gersting book does not even mention the Sets of the Algebraic and Transcendental Numbers. However, here is a list of all of the sets of numbers you may run across in this course:

The Empty Set and the Universal Set

 * The empty set is the set that contains no elements, and is denoted as:




 * The empty set is sometimes also given as an empty set of brackets:




 * The universal set is a set that contains all elements under discussion.

Basic Set Comparisons
Sets A and B are equal if every element of A is also an element of B and every element of B is also an element of A:



Set A is a subset of B if every element of A is also an element of B:



Set A is a superset of B if every element of B is also an element of A:



Set A is a proper subset of B if A is a subset B and A is not equal to B:



Set A is a proper superset of B if A is a superset of B and A is not equal to B:



Basic Set Operations
The union of sets A and B is the set of elements that are a member of A or a member of B:



The intersection of sets A and B is the set of elements that are both a member of A and a member of B:



The difference of set A and set B is the set of elements that are a member of set A and are not a member of set B:



The complement of set A is the set of all elements that are not a member of A, or, alternatively, the difference of the universal set and A:



Basic Set Properties
The cardinality of a set is the number or quantity of elements the sets contains, although a formal definition for cardinality is rather complex. Cardinality is usually denoted using the following notation: