The Empty Set

The empty set is the unique set having no elements such that its cardinality is 0.

Definition

A set which does not contain any element is called the empty set or the null set or the void set.

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 10 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 10 doors. So, the set containing the van with 10 doors is an empty set.

Cardinality of Empty Set

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.

How To Represent an Empty Set?

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

(i) Let $A=\{x:1<x<2,x$ is a natural number$\}$ Then $A$ is the empty set, because there is no natural number between $1$ and $2$.

(ii) $D=\{x:x^{2}=4,x$ is odd $\}$.Then $D$ is the empty set, because the equation $x^{2}=4$ is not satisfied by any odd value of $x$