 In mathematics, a set is a well-defined collection of elements or objects. Finite and Infinite Sets are defined by the number of elements in a set. However, there is another sort of set known as the empty set. The empty set is the only set that has no elements and so has a cardinality of 0. In most textbooks and publications, the empty set is referred to as the “null set.”

Unfortunately, in the context of measure theory, the null set is a different concept. It symbolises a set of measure zero, indicating that this set is not empty. As a result, the Empty Set is also known as the void set.

On that note, let’s learn the mathematical definition of an empty set, symbol, examples and properties in detail.

## Empty Set Definition

The empty set, null set, or void set is a set that does not include any elements. For example, the collection of outcomes for rolling a die and getting a number greater than 6. As we all know, the results of a die roll are 1, 2, 3, 4, 5, and 6. As a result, the set containing integers greater than 6 here will be { }. This indicates that there will be no elements and is referred to as the empty set.

The empty set is represented by the symbols “Φ” and “φ” or { }.

## Empty Set Examples

Let’s have a look at some empty set examples below.

• Consider the set A = {x: 3 x 4, where x is a whole number}, and this set A is the empty set because there is no whole number between 3 and 4.
• Assume B = {x: x is a composite number less than 4}, hence B is an empty set because 4 is the smallest composite number.
• The set C = {x: x2 – 3 = 0, where x is a rational number}, is the empty set because no rational value of x can satisfy the equation x2 – 3 = 0.
• The set D ={ x: x3 = 27, x is even} is the empty set, because no even value of x can satisfy the equation x3 = 27.

## Properties of Empty Set

Let’s discuss some of the most important properties of an empty set:

• Cardinality: The number of elements in the empty set, or its cardinality, is zero, |φ| = 0.
• The empty set is a subset of every set: the empty set is a subset of every set X, i.e. φ ⊆ X; ∀ X.
• The subset of an empty set: The only subset of an empty set is the empty set itself, i.e. X ⊆ φ ⇒ X = φ.
• A cartesian product containing an empty set: The empty set is the Cartesian product of a set B and the empty set is always an empty set, i.e. B × φ = φ; ∀ B.
• The empty set’s power set: The empty set’s power set is the set that only contains the empty set, i.e. 2φ = {φ}.
• Union with Empty set: The set X is the union of A and the empty set, i.e. X ⋃ φ = X; ∀ X.
• Intersection with Empty: The intersection of A with the empty set is the empty set once more, i.e. X ⋂ φ = φ; ∀ X.

## Empty Set is Finite or Infinite

Because its cardinality is defined and equal to zero, an empty set is a finite set. As we know, an infinite set has an infinite number of items, i.e. its cardinality is ∞ or is not specified, whereas a finite set has a countable number of components.