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Algebra Foundations

You will learn how to determine empty, universal, finite, infinite, equal, and equivalent sets using Venn diagrams, the listing method, and/or set builder notation.

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After completing this tutorial, you will be able to complete the following:

- Define the concept of sets.
- Define empty, universal, finite, and infinite sets.
- Determine equal and equivalent sets, when the sets are given with different representations.
- Identify the elements and number of elements of sets, when the sets are given with different representations.

This Activity Object introduces learners to the basics and properties of sets.

Sets and Their Elements

A set is a group of members or elements with one or more common characteristics. Sets are named with capital letters. For example, your desktop may hold a pencil, pen, and pad of paper. This "set of items on your desk" can be named set A. Another desk may have a pen, pad of paper, notebook, and eraser. This group of objects can be named set B.

Each object in a set is called a member or element of the set. Using set notation, you can say the pencil is an element of set A like this:

You can count the number of elements in a set. Using the listing method of representation, you can see that set A has three elements and set B has four:

A = {pencil, pen, pad of paper}

B = {pencil, pad of paper, notebook, eraser}

This can also be represented using n for number:

n(A) = 3

n(B) = 4

Representations of Sets

Sets are commonly represented in three different ways.

1. The listing method is used to represent sets with limited elements:

A = {pencil, pen, pad of paper}

2. Set builder notation is another representation. Set builder notation is usually used to represent a set which has many elements:

A = {x | x is a pencil, pen, pad of paper}

This representation is stated as "the set of all x, such that x contains the elements pencil, pen, and pad of paper."

The set "all integers from 3 to 933" is a very large set and can be most easily represented in set builder notation as follows:

A = {3, 4, 5,..., 931, 932, 933}

3. Use a Venn diagram to illustrate sets that do not have many elements. Venn diagrams are excellent for illustrating intersections of sets (the common elements of each set). In the following illustration, you can see the elements of set A, the elements of set B, and the elements in both sets.

Empty, Universal, Finite, and Infinite Sets

1. An empty set has no elements. For example, suppose you consider the number of live zoo animals in the room. If there are none, the set is empty. Represent an empty set with an empty Venn diagram or one of the following notations:

n(C) = 0

C = { }

C = Æ

2. A universal set is the set of all elements under consideration. For example, if the set includes all the elements seen on both desks, the following Venn diagram depicts the universal set (U).

3. A finite set has a given number of elements. The elements on a desktop make up a finite set. Other finite sets representations follow:

A = {10, 11, 12,..., 18, 19, 20}

N(A) = 92

4. An infinite set is a set with infinitely many elements. It is not possible to list all the elements. Infinite sets can be represented as follows:

A = {...-2, -1, 0, 1, 2,...}

Approximate Time | 30 Minutes |

Pre-requisite Concepts | Students should know the concepts of integers and infinity. |

Course | Algebra Foundations |

Type of Tutorial | Concept Development |

Key Vocabulary | element, empty set, equal sets |