Set Theory

A discussion of set theory requires the use of some standard symbols. Unfortunately, these symbols do not display properly on some computers. They should work on any MS Windows based system. In the table of symbols below, if they are not displayed properly, you will not be able to read this section easily. We apologize for that. If you are using Netscape under Unix here are some instructions on how to get the symbol font working. It also has some advice for Macintosh users.

Symbol	Name
Æ	the empty set
Î	element of
Ï	not an element of
Í	contained in
Ê	contains
Ë	not contained in
Ç	intersection
È	union

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Terminology

A set is any collection of objects. Examples.

The empty set is the set containing no elements, denoted by Æ.

If A is a set and x is a member of A, we say x is an element of A and denote this by x Î A.

If A and B are sets and every element of A is also an element of B (that is, x Î A implies x Î B), then we say A is a subset of B or A is contained in B and we denote this by A Í B.

The intersection of two sets is the elements they have in common. For example, if A={1, 2, 3, 4} and B={2, 4, 6, 8}, then A Ç B = {2, 4}.

The union of two sets is the set of elements that are in at least one of the two sets. For example, if A={1, 2, 3, 4} and B={2, 4, 6, 8} then A È B = {1, 2, 3, 4, 6, 8}.

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Notation:

As you may have surmised from the terminology section, the following conventions for notation are used in set theory:

• A, B, X: capital letters denote sets
• a, b, x: lowercase letters denote elements of sets
• { }: curly braces denote a list of elements in a set

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Venn Diagrams

When working with sets, it is often easier to visualise concepts with a picture. We can do this with Venn Diagrams. A set is drawn as a geometric area (e.g. circle, rectangle) and shading is used to indicate a specific portion of the set or sets. Here are some Venn Diagrams that illustrate the concepts we have already discussed:

Element	Subset	Intersection	Union
x Î A	A Í X, B Í X	A Ç B	A È B

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Set Operations

Some mathematical operations can be applied to sets. We will discuss equality and subtraction.

Two sets A and B are equal, written A = B, when they have precisely the same elements. This is the same as saying that A È B = A Ç B. Look at the definitions and Venn Diagrams above to understand why this must be the case.

The difference of two sets is defined as: A - B = {x| x Î A and x Ï B}. This mathematical notation might look confusing, but we can translate each part into words (see below). The Venn diagram makes this concept more understandable.

A - B The difference of A and B = is { x the set of elements x | such that x Î A x is an element of the set A and x Ï B} and x is not an element of the set B

Set Difference A - B

From the definition of set difference, we can derive the concept of a set complement. If A is a subset of X, then the complement of A in X is the set A^c = X - A. This is illustrated with a Venn Diagram below:

Set Complement

Ac

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Subsets

The basic definition of subsets is in the terminology section. Here are four further theorems about subsets:

• For any set A, Æ Í A
• For any set A, A Í A
• If A, B, C are sets with A Í B and B Í C, then A Í C.
• A = B if and only if A Í B and B Í A.

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