**An Introduction to Set Theory**

A **set** is a collection of objects. We group numbers into sets for various reasons, including one of the most common groupings: domains and ranges of functions. In order to work with sets of numbers effectively, we need to understand a bit of the basic theory of sets.

There are often many ways to define a set. The following three examples consist of the same set defined in two different ways.

a) A = .

This is a simple list of all members of set A.

b) Set A can also be described as the set of all even integers larger than 0 and

less than 20.

c) We can further describe Set A as {x | 2 ≤ x ≤ 18 and x is even.} This is called

set-builder notation and it uses a more formal description of the objects in the

set. This notation would be read, “The set of all values of x such that x is

greater than or equal to 2 and less than or equal to 18 and x is even.”

A capital letter is often used to denote a set.

For the following examples:

A =

B =

C = .

The objects in a set are called **elements** of the set.

The number 6 is an element of A. This is written in symbols as .

The number 5 is less than 20, but it is odd. Therefore, 5 is not an element of A. This is written in symbols as .

Set B is a **subset** of set A if and only if every element of B is also an element of A. Even if A and B were identical sets, then B would still be considered to be a subset of A.

Using our definitions of A, B, and C, above, set B is a subset of A.

This is written in symbols as .

The set C is not a subset of A because it contains elements not in A (namely, 13 and 15).

This is written in symbols as .

Set B is a **proper subset **of set A if and only if every element in B is an element of A **AND** A contains at least one element that is not in B. The set B is a proper subset of A, since 2, 4, 6, 8, 10 are elements of A and are not elements of B, and also every element of B is in A.

This is written in symbols as .

The **union** of sets A and B, written A U B, is the set of all elements contained in A or contained in B.

A U B = {2, 4, 6, 8, 10, 12, 14, 16, 18}

The **intersection** of sets A and B, written A ∩ B, is the set of all elements that A and B have in common.

A ∩ B = {12, 14, 16, 18}

The **difference** of sets A and B is defined to be the set of elements that are in A but not in B.

A - B = {2, 4, 6, 8, 10}

The **empty set** is the set without any elements. It is written as

The **Cartesian product** of sets A and B is the set of all ordered pairs (a, b) where a is in set A and b is in set B.

Let A = and B =

Then

A **Venn diagram** is a diagram which illustrates the relationships between sets. The Venn diagram below represents the intersection of sets A and B (e.g., A U B). The shaded region represents the set of elements that belong to set A and to set B.