# Subsets in Math: Definition & Examples

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• 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

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Lesson Transcript
Instructor: David Liano
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:

• 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.

## 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.

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