Finite Set: Definition & Overview

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Geometric Sequence: Formula & Examples

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 What is a Set?
  • 0:54 What is a Finite Set?
  • 1:23 Examples of Finite Sets
  • 2:52 Examples of Infinite Sets
  • 3:36 Sets as Elements of Other Sets
  • 4:42 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

After completing this lesson, you will be able to define what a finite set is and give examples of a finite set. You will also be able to explain the difference between a finite set and an infinite set.

What Is a Set?

Before we define a finite set, we need to establish what a set is. A set is a collection of elements or objects. Each element can be distinguished from other elements in the set. An example of a set would be all the prime numbers less than 20. Let's call this set A. Using proper set notation, we can write A as follows:

A = {2, 3, 5, 7, 11, 13, 17, 19}

Braces are typically used to enclose the elements of a set.

The cardinality of a set is the number of elements in the set. For instance, set A as defined earlier has a cardinality of 8 because it has eight elements. Using proper notation, we say that |A| = 8. The vertical bars indicate the cardinality of A.

What Is a Finite Set?

A finite set is a set that has a cardinality that equals a natural number (1, 2, 3…). Let's say that B represents a set. If |B| = n such that n is a natural number, then B is a finite set.

The empty set is also a finite set. The empty set has no elements and is denoted by the symbol Ø or by a pair of braces { }. The cardinality of the empty set is 0 (|Ø| = 0).

Examples of Finite Sets

The set described at the start of this lesson is an example of a finite set. Set A was defined as the prime numbers less than 20. There are eight prime numbers less than 20, so A had eight elements or a cardinality of 8.

Let's define a new set as follows:

B = (a, e, i, o, u)

B is the set of vowels in the English alphabet. B is a finite set because it has five elements or a cardinality of 5.

Let's define set C as the set of natural numbers less than 10. We can show C as follows:

C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

We can also show C using an alternative form of set notation as shown in the following figure, which we will call Figure 1. The boldface capital N is often used to indicate the set of natural numbers. We can read the notation in Figure 1 as all elements x in the set of natural numbers such that x is less than 10.

Finite sets do not have to contain numbers and/or individual letters. For instance, the continents of the world can be a set as shown next:

{Africa, Antarctica, Asia, Australia, Europe, North America, South America}

This set of continents has a cardinality of 7.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account