The Empty Set in Math: Definition & Symbol

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  • 0:01 Set
  • 0:36 The Empty Set
  • 1:01 Cardinality of the Empty Set
  • 1:38 Empty Set as a Solution Set
  • 2:46 Empty Set as a Subset
  • 3:32 Empty Set & the Power Set
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Lesson Transcript
Instructor: David Liano

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

After viewing this lesson, you should know how to apply the empty set within mathematics topics. You should also know how to explain the empty set in words and to write the empty set in symbols.

Definition: Set

Before we define the empty set, we need to establish what a set is. A set is a collection of distinct elements or objects. Each element is discernible from the other elements of the set. In other words, we need to be able to distinguish one element from another.

An example of a set would be all the natural, or counting, numbers less than 5. Let's call this Set A. Using proper set notation, we can write Set A as follows: A = {1, 2, 3, 4}. Braces are typically used to enclose the elements of a set.

Definition: The Empty Set

The empty set is a set with no elements. We can use braces to show the empty set: { }. Alternatively, this symbol, Ø, is often used to show the empty set. As the picture shows, the two symbols mean the same thing. We can think of the empty set as a box with nothing in it. The empty set exists like the box exists. We can think of the braces as representing the container.


Cardinality of the Empty Set

The cardinality of a set is the number of elements in the set. Set A defined earlier as the counting numbers less than 5 has a cardinality of 4 because it has four elements: the numbers 1, 2, 3, and 4.

Using set notation, we write it this way: |A| = 4. The vertical bars on either side of 'A' indicate the cardinality of Set A. The cardinality of the empty set is 0 because the empty set has no elements. In set notation, we can write |Ø| = 0.

The Empty Set as a Solution Set

If a problem has no solution, the solution can be represented by the empty set. For instance, let's state this problem:

'Name all the states in the United States that begin with the letter Z.'

There are no states in the United States that begin with the letter Z. Therefore, the solution to this problem is the empty set: Ø. However, we need to distinguish between the empty set and the number zero as an answer. Let's reword the previous problem as follows:

'How many states in the United States begin with the letter Z?'

The answer to this question is 0. Using set notation, we would write the solution as {0}. This solution contains one element, the number 0, so its cardinality is 1. It is not empty! Let's look at a problem in algebra. What values of x make the following equation true: x + 5 = x + 3?

We should see that there are no possible values of x that will make this equation true. The left side of the equation will always be larger than the right side of the equation. Therefore, the solution, or the domain of x, is the empty set.

The Empty Set as a Subset

The empty set is a subset of every set. Let's define subset. A set is a subset of another set if every element of the set is also an element of the other set. Set A is a subset of Set B if every element of A is also an element of B.

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