Null Set: Definition

Null Set: Definition & Example

The null set is the set that contains no elements. In this lesson. learn how the null set arises naturally when you place certain restrictions on sets, and explore some very useful properties in set theory.

What Is the Null Set?

The null set, also referred to as the empty set, is the set that contains no elements. For example, suppose somebody asked you to find the set of all senior citizens who are less than five years old. Clearly, there are no senior citizens under five because you have to be much older than five to be considered a senior citizen! Therefore, your set contains no elements and is the null set. Another example of the null set is the set of all even numbers that are also odd. Clearly a number cannot be both odd and even, so there are no elements in this set.

Null Set Notation

There are two pieces of notation that are most commonly used. The first is {} (curly brackets). The second is Ø (slashed zero). Since the most common way to notate a set is by placing the elements inside of a pair of curly braces, the first notation seems to be the most natural. However, the slashed zero was first introduced in 1939, by a group of mathematicians. Since then, it has become the more widely used notation.

Properties of the Null Set

There are five properties of the null set:

1. For any set A, the null set is a subset of A

Ø ? A

Notice that this is true for every set, even the null set!

Therefore, if you were to list the subsets of each set, the null set would appear on every single list.

2. For any set A, the null set is a union of A

Ø ? A = A

Since the union of two sets is the set of all elements in both sets and the null set has no elements, all of the elements in both sets are only the elements in A.
3. For any set A, the intersection of A with the null set is the null set

   Ø n A = Ø

   Since the intersection of two sets is the set of all elements in both sets and the null set has no elements, they have no elements in common

4. The only subset of the null set is the null set itself

This relates back to the first property since the null set is the subset of every set. Since the null set contains no elements, we can't create any other subset of the null set besides the null set.

5. The cardinality of the null set is 0

Since the cardinality of a set is the number of elements in the set, and the null set has no elements, its cardinality must be 0.

Misconceptions of Null Set Notational

Be very careful when determining if a set is the null set. For example, :

{Ø}

This is not the null set. It's the set containing the null set! So this set actually has one element, namely the null set. Since the null set has no elements but this has one element, they cannot be the same. Similarly, look at this set:

{Ø, {Ø}}

This is also not the null set. This set contains two elements, namely the null set and the set containing the null set and, therefore, cannot be the null set.

Lesson Summary

The null set is the set that contains no elements. It is denoted either by {} (curly brackets) or Ø (slashed zero). It has five properties:

1. For any set A, the null set is a subset of A
2. For any set A, the null set is a union of A
3. For any set A, the intersection of A with the null set is the null set
4. The only subset of the null set is the null set itself
5. The cardinality of the null set is 0