Empty Set (Null Set)

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = {}. It is also called a void set or null set.

Empty sets are considered to be unique when compared to other sets. In this article, let's learn about empty sets, their definition, properties, and examples.

Empty set is a set without any elements. The set without any elements has to be also represented and is represented as an empty set of a null set. The empty set is denoted by ∅ = { }.  Empty sets are used to simplify the calculations and are most often used while classifying all the odd items or those items that are considered to be rare. Here are some examples where an empty set can be used for the classification of odd elements. For instance, a month with 33 days, a week with 2 Tuesdays, or a cat with five legs. Let's go ahead and learn the definition of empty sets.

Definition of Empty Sets

A set can be defined as an empty set or a null set if it doesn't contain any elements. In set theory, an empty set may be used to classify a whole number between 6 and 7. Since this example does not have any definite answer, it can be represented using an empty set or a null set.

Let's consider the following examples where we need to determine if the given sets are empty sets.

a.) X = {x : x is a prime number and 14<x<16}

We will consider the set of prime numbers as A. Thus, A = {2, 3, 5, 7, 11, 13, 17…}. Since there are no prime numbers between 14 and 16, we can conclude that the X is an empty set.

b.) Number of vans with 12 doors.

In real life, unless there is a situation where a van manufacturing company creates a particular model, it’s impossible to find a van that has 12 doors. So, the set containing the van with 12 doors is an empty set.

Empty Set Symbol

An empty set is represented as {}, containing no element at all. It is also represented using the symbol '∅' (read as 'phi').

Cardinality of Empty Set

Empty sets are considered to be unique sets in set theory and thus, they also possess a unique cardinality. Cardinality can be defined as the size of the set or the total number of elements that are present in a set. As empty sets do not contain any elements, we can say that their cardinality is zero.

In set theory, empty sets are represented by using the empty curly brackets { } that are generally used to denote sets. However, since empty sets are unique kinds of sets, these sets can also be represented by using the special character ∅. Let us consider an example of an empty set A containing multiples of 5 between 6 and 8. Since there are no multiples of 5 between 6 and 8, so the given set is an empty set. Set A = {y: y is a multiple of 5 and 6<y<8}

• The notation of this empty set is: A = { }.
• The same empty set A can also be represented using the following notations: Empty set = { } or, X = { }.
• We can use the special symbol ∅ to represent an empty set.

Most often the concepts of zero sets and empty sets are considered to be the same and even the terms(zero set and empty set) are used interchangeably. This misconception can be avoided by having a clear understanding of these two sets.

This difference between the zero set and the empty set shows why the empty set is considered as unique as it has an element-less characteristic. Thus, these two sets are distinct from one another. Let's consider an example to understand the difference better.

• Consider a set Y = {0} and a set X = {x : x is an odd number and 5<x<7}.
• Now, to differentiate between the two sets, we need to first simplify them.
• We can see that there is no odd number between 5 and 7, hence set Z is an empty set and it can be written as: X = { } or ∅ = X.

It is evident that set X is an empty set, whereas set Y is a zero set.

Venn Diagrams are considered the most effective tool to represent the relationships between sets, especially finite sets. We can represent an empty set through a Venn diagram. The below-given image illustrates the relation as follows:

Consider a set X = {1, 3, 5} and a set Y = {2, 4, 6}

We can see from the above-given Venn diagram that there are no common elements that are intersecting elements between the two sets X and Y, hence we can conclude that the intersection between these two sets is empty. Thus, X ∩ Y = ∅.

Empty sets are said to play an important role in the classification of objects that are unique and odd. In addition to making the classification easier, empty sets are used to simplify the calculations. Here are some of the important properties of empty sets:

Property 1: Subset of any Set

An empty set can be considered as the subset of any given set. For any finite or infinite set X, if we exclude all the possible subsets of set X, then we can always include an empty set in this.

For example,

• Consider a finite set X = {1, 3, 5}.
• All the possible subsets of this set X can be written as: X = ∅, X = {1}, X = {3}, X = {5}, X = {1,3}, X = {3, 5}, X = {1,5}.
• As you can see, the empty set has also been included along with the other subsets because of this property.

Property 2: Union with an Empty Set

The set operation of union of sets between any set and an empty set will always result in the set itself. For any finite or infinite set X, the union of this set X with an empty set is X U ∅ = X. Since an empty set does not contain any elements of its own, the union between an empty set and any set X produces the same set X as a result.

For example,

• Consider a set X = {1, 2, 3, 4}.
• The union of the given set X with an empty set can be shown as X U ∅ = {1, 2, 3, 4} U { }. Thus, A U ∅ = {1, 2, 3, 4}

Property 3: Intersection with an Empty Set

The set operation of intersection of sets between any set and an empty set will always result in the set itself. For any finite or infinite set X, the union of this set X with an empty set is X ∩ ∅ = X. Since an empty set does not contain any elements of its own, there will not be any common element between any non-empty set and an empty set.

For example,

• Consider a set X = {2, 4, 6}.
• The union of the given set X with an empty set can be shown as X ∩ ∅ = {2, 4, 6}

Property 4: Cardinality of Empty Set

Cardinality refers to the size of the set. In other words, it is the total number of elements in the given set. An empty set contains no elements. Thus, it has cardinality equal to zero.

For example,

• Consider a set X = {x: x is an odd multiple of 2}.
• Odd numbers are the ones that are not divisible by 2. Thus, there are no odd multiples of 2. There is no element in set A or set A = {}. Therefore, the cardinality of set A = 0.

Property 5: Cartesian Product of Empty Set

The Cartesian product of a set and an empty set, say set A and an empty set = A × φ = φ, ∀ A. This further implies that the cartesian product of a set with an empty set is always an empty set.

• Consider a non empty set A = {1, 2, 3, 4} and an empty set = {}.
• Their cartesian product =A × φ = φ

Related Articles on Empty Set

Check out the following pages related to the empty set

• Finite and Infinite Sets
• Sets Formula
• Intersection of Sets
• Operation on Sets

Important Notes on Empty Set

Here is a list of a few important points related to an empty set

• A set can be defined as an empty set or a null set if it doesn't contain any elements.
• Empty sets are also called void sets or null sets.
• The union operation between any set and an empty set will always result in the set itself.
• Since the cardinality of an empty set is defined and is equal to 0, therefore, an empty set is a finite set.

What Is an Empty Set in Math?

A set can be defined as an empty set or a null set if it doesn't contain any elements. In set theory, an empty set may be used to classify a whole number between 6 and 7.

What Are the Properties of an Empty Set?

Here are some of the important properties of empty sets:

• An empty set can be considered as the subset of any given set.
• The union operation between any set and an empty set will always result in the set itself.
• The intersection operation between any set and an empty set will always result in the set itself.

Is Empty a subset of every set?

An empty set is characterized by the property which states that it has no elements at all. This further means that every element in the empty set (actually none) belongs to any set. Hence, an empty set is a subset of every set.

Is Ø an Empty Set?

Yes, ∅ is an empty set. An empty set does not contain any elements. Most often the special symbol ∅ is used to denote an empty set.

What Is the Power Set of an Empty Set?

An empty set or a null set does not contain any element. Therefore, the power set of the empty set is an empty set only.

Is 0 in an Empty Set?

No, 0 is not in the null set as an empty set does not contain any elements in it. Nothing and zero cannot be considered as the same thing.

Is an Empty Set Finite or Infinite?

Yes, an empty set can be considered as a finite set since its cardinality is 0.

What Kind Of Sets Results In Their Difference As An Empty Sets? 

The sets which are equal sets result in their difference as empty sets. The equal sets have the same elements and the number of elements is also equal. Hence the difference of equal sets is an empty set.

Why Is the Empty Set Important?

Empty sets are said to play an important role in the classification of objects that are unique and odd. In addition to making the classification easier, empty sets are used to simplify the calculations. Empty sets are considered to be an important component in number theory as the natural numbers all are formed from the empty set. Without an empty set, natural numbers cannot exist mathematically.

How Do You Represent an Empty Set?

In set theory, empty sets are represented by using the empty curly brackets { } that are generally used to denote sets. However, since empty sets are unique kinds of sets, these sets can also be represented by using a special character ∅.