Fri. Jun 2nd, 2023

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.

By Manali