Subsets in Math: Definition & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Understanding Bar Graphs and Pie Charts

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

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:01 Definition of a Set
  • 1:40 Definition of Subsets
  • 2:41 Identifying Subsets
  • 4:41 Subsets of Number Systems
  • 5:42 Empty Set & Power Set
  • 6:47 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
David Liano

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

Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

After completing this lesson, you should be able to define the concept of subsets in math. You should also be ready to write subsets using proper set notation. Finally, you should be able to identify sets that are subsets of other sets.

Definition of a Set

Before we define subset, we need to refresh ourselves on what a set is. A set is a collection of elements or objects. These elements are usually related in some way, but this is not necessary. However, each element needs to be discernible from the other elements of the set. In other words, we need to be able to distinguish one element from another. It's probably best to show some examples of sets.

'All the states of the United States' is an example of a set with 50 elements. But we could also make a set of only two states.

  • For example, we could create a set that has only Nebraska and Ohio as its elements: A = {Nebraska, Ohio}.
  • All the letters of the English alphabet is an example of a set with 26 elements. But again, we could create a set that has only three letters: B = {d, g, z}.
  • We could even make a set that includes both states and letters. Let's call this set C: C = {d, g, z, Nebraska, Ohio}.

The examples above used appropriate set notation. Braces ({}) are usually used when writing down a set. It is often common to use capital letters to name a set.

One more thing before we move on: The elements of a set can be written in any order and can be listed more than once. We could also write the set C as C = {g, d, Nebraska, z, d, Ohio, Ohio}. We did not change set C. It still has the same five elements.

Definition of Subsets

Let's use sets A, B and C again as defined above and listed again here:

  • A = {Nebraska, Ohio}
  • B = {d, g, z}
  • C = {d, g, z, Nebraska, Ohio}

We can say that A is a subset of C because all the elements of A are also elements of C. A subset is a set made up of components of another set.


Set A is more specifically a proper subset of set C because A does not equal C. In other words, there are some elements in C that are not in A. A proper subset is a subset that is not equal to the set it belongs to. Some textbooks or websites will use this notation to specify a proper subset (note that the underscore is removed). In this lesson, the first figure we showed will be used for all subsets.


We can also show the relationship between A and C in a Venn diagram.

venn diagram

Identifying Subsets

Now let's talk about identifying subsets. This part of the lesson gets a little tricky. First, we need to accept that there can be sets within sets. This means that sets can be elements of other sets. Let's look at this example:

D = {2, {2}, 3, 4, {7}, {11, 12} }

In set D, there are six elements. Three of the elements are sets of one number or of multiple numbers: {2}, {7} and {11, 12}. We can distinguish these elements as sets because of the brackets. The other three elements are individual numbers: 2, 3 and 4.

Elements 2 and {2} are not the same because {2} is a set and 2 is not a set. In addition, the numbers 11 and 12 are not elements of D, but the set {11, 12} is an element of D. This distinction between elements and sets is straightforward, but often is a difficult rule to apply. It would probably be beneficial to review this part of the lesson more than once.

Now, let's identify some subsets of D. We need to recall the definition of a subset: A is a subset of B if all the elements of A are elements of B. If set F = {3, {7} }, then F is a subset of D because all the elements of F are elements of D. Similarly, if set G = {2, {2}, {11, 12}, 4}, then G is a subset of D because all the elements of G are elements of D.

Now, let's identify a non-example. If set H = {3, 7}, then H is not a subset of D. The set containing the number 7 is an element of D, but the number 7 is not an element of D. Note the difference in notation between F and H.

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

Finding the Number of Sets in a Power Set Activity


  • The empty set is a subset of all sets.
  • The power set of a set, S, is the set of all subsets of S.


  1. Consider the empty set, S0 = { }. Determine how many subsets that the empty set, S0, has. In other words, determine the number of sets in the power set of the empty set, S0.
  2. Consider a set with 1 element, S1 = {a}. Determine how many subsets that S1 has. In other words, determine the number of sets in the power set of S1.
  3. Consider a set with 2 elements, S2 = {a,b}. Determine how many subsets that that S2 has by determining the number of subsets with no elements, the number of subsets with 1 element, and the number of subsets with 2 elements, and then add up the total number of subsets. In other words, determine the number of sets in the power set of S2.
  4. Repeat this for a set with 3 elements, S3 = {a,b,c), and for a set with 4 elements, S4 = {a,b,c,d}. In other words, find the number of sets in the power set of S3 and in the power set of S4.


  1. Do you notice any patterns emerging in terms of the number of sets in the power set of a given set?
  2. Use these patterns to determine a formula for the number of sets in the power set of a set, Sn, with n elements.
  3. Use this formula to determine the number of sets in the power set of a set with 8 elements (S8), a set with 15 elements (S15), and a set with 20 elements (S20).

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