Empty Set
An empty set is a type of set that contains no elements at all. It is also called the null set and is denoted by the empty set symbol $\emptyset$ or $\{\}$. For example, the set of natural numbers less than $0$ is an empty set because no such number exists. In simple terms, an empty set is a set with zero elements.
- What is an Empty Set?
- Venn Diagram of Empty Set
- Properties of Empty Set
- Is Empty Set Finite or Infinite?
- What is the difference between Zero Set and Empty Set?
- Empty Set Examples
This article will help you understand what is empty set, see examples of empty set, learn its properties, and also find out whether the empty set is finite or infinite in mathematics. If you’re wondering which of the following sets is empty set, keep reading to explore more with simple explanations and real-life examples.
What is an Empty Set?
An empty set is a set that contains no elements. It is also called the null set or void set. In other words, if a set has nothing inside it, it is considered an empty set.
The symbol for empty set is either $\emptyset$ or $\{\}$. Is $Ø$ an empty set ? Yes it is empty set or null set.
Empty Set Notation
An empty set is usually written in one of the following ways:
$\emptyset$
$\{\}$
Both represent a set with zero elements.
Examples of Empty Set
Here are some common and easy-to-understand examples of empty sets:
Set of natural numbers less than 0
$A = \{x \in \mathbb{N} \mid x < 0\}$
Since natural numbers start from 1, this set has no elements. So, $A = \emptyset$.Set of real numbers whose square is negative
$B = \{x \in \mathbb{R} \mid x^2 + 1 = 0\}$
No real number satisfies this equation because $x^2$ is always non-negative. Hence, $B = \emptyset$.Set of months with 32 days
$C = \{\text{months with 32 days}\}$
No month has 32 days, so this is also an empty set: $C = \emptyset$.
Note that: The cardinality (number of elements) of an empty set is 0, i.e.,
$n(\emptyset) = 0$
Venn Diagram of Empty Set
Venn diagrams are useful for showing relationships between sets. An empty set can be represented by leaving the region of the set blank, showing it contains no elements.
Consider a set $P =\{1, 0, 5\}$ and a set $Q = \{2, 8, 6\}$
We can see that there are no common elements between the two sets P and Q, hence the intersection between these two sets is empty. So, P ∩ Q = ∅.
Properties of Empty Set
The empty set, also called the null set, plays a vital role in set theory. It has unique properties that help define operations and relationships between sets. Below are the most important properties of the empty set, explained with correct notation and symbols:
Property | Description | Mathematical Representation |
Empty Set Symbol | The empty set is represented by the Greek letter phi or empty braces. | $\phi$ or $\{\}$ |
Cardinality | The number of elements in the empty set is zero. | 0 |
Subset of Every Set | The empty set is a subset of every set. | $\phi \subseteq A,\ \forall A$ |
Subset of Empty Set | The only subset of an empty set is the empty set itself. | $A \subseteq \phi \Rightarrow A = \phi$ |
Cartesian Product | The Cartesian product of any set with the empty set is the empty set. | $A \times \phi = \phi,\ \forall A$ |
Power Set | The power set of the empty set contains only the empty set. | $2^{\phi} = {\phi}$ |
Union with Empty Set | Union with the empty set gives the set itself. | $A \cup \phi = A,\ \forall A$ |
Intersection with Empty Set | Intersection with the empty set gives the empty set. | $A \cap \phi = \phi,\ \forall A$ |
Remarks on Empty Set
$\phi$ is called the null set.
$\phi$ is unique, meaning there is only one empty set.
$\phi$ is a subset of every set.
$\phi$ is not written within braces; that is, ${\phi}$ is not the empty set.
${0}$ is not an empty set, as it contains the element $0$ (zero).
Example:
${\x \mid x \in \mathbb{N},\ 4 < x < 5\} = \phi$
This set has no natural number between 4 and 5, so it is an empty set.
Is Empty Set Finite or Infinite?
Is Empty set countable? Empty set doesn't have any element to count. However, the cardinality of an empty set is $0$. We can check the cardinality of a set by finding its cardinal number.
An empty set is a finite set because it has a defined and countable number of elements—specifically, zero. Its cardinality is $0$, which means the number of elements in the set is known.
In contrast, a set is called infinite if it has an uncountable or undefined number of elements, i.e., its cardinality is $\infty$. Since the empty set has no elements, and $0$ is a definite number, it is always finite.
What is the difference between Zero Set and Empty Set?
Though they may sound similar, the zero set and empty set are different. This section explains how they differ based on elements, notation, and meaning in set theory.
Aspect | Empty Set | Zero Set |
Definition | A set with no elements. | A set that contains only the number zero. |
Notation | $\emptyset$ or ${}$ | ${0}$ |
Cardinality | $0$ (no elements) | $1$ (one element) |
Contains 0? | Does not contain $0$ or any other element. | Contains only the element $0$. |
Type of Set | Null/empty set | Singleton set |
Example | ${x \in \mathbb{N} \mid x < 0}$ | ${x \in \mathbb{R} \mid x = 0}$ |
Empty Set Examples
Example 1: Which of the following is NOT true?
1) Equivalent sets can be equal.
2) Equal sets are equivalent.
3) Equivalent sets are equal.
4) None of these
Solution: In this Question,
Equivalent sets may or may not be equal sets but equal sets always have the same number of elements and hence equal sets are always equivalent.
Hence, the answer is the option 3.
Example 2: Which of the following sets is empty set?
1) $A=\{x: x$ is an even prime number $\}$
2) $B=\{x: x$ is an even number divisible by $3\}$
3) $C=\{x: x$ is an odd integer divisible by $6\}$
4) $D=\{x: x$ is an odd prime number $\}$
Solution: In this Question,
Option $1=\{2\}$, Option $2=\{6,12,18, \ldots$.$\}$ , Option $4=\{3,5,7, \ldots\}$
For option 3 , as there are no odd numbers divisible by $6$ , so it is an empty set Hence, the answer is the option 3.
Example 3: Which of the following is not an empty set?
1) Real roots of $x^2+2 x+3=0$
2) imaginary roots of $x^2+3 x+2=0$
3) Real roots of $x^2+x+1=0$
4) Real roots of $x^4-1=0$
Solution: Option (1) has imaginary roots as Discriminant $<0$. So, there is no real root, and the set is empty.
Option (2) has real roots as Discriminant $>0$. So, there is no imaginary root, and the set is empty.
Option (3) has imaginary roots as Discriminant $<0$. So, there is no real root, and the set is empty.
Option (4): $x^4-1=0 \Rightarrow\left(x^2-1\right)\left(x^2+1\right)=0$, which gives $x= \pm 1$.
Hence, it has $2$ real roots. So, it is not empty.
Hence, the answer is the option 4.
Example 4: Which of the following is empty set?
1) $\left\{x\right.$ : $x$ is a real number and $\left.x^2-1=0\right\}$
2) $\left\{x: x\right.$ is a real number and $\left.x^2+1=0\right\}$
3) $\left\{x\right.$ : $x$ is a real number and $\left.x^2-9=0\right\}$
4) $\left\{x\right.$ : $x$ is a real number and $\left.x^2+2=0\right\}$
Solution: Empty Set- A set which does not contain any element is called the empty set or the null set or the void set.
Wherein e.g. $\{1<x<2, x$ is a natural number $\}$
Since $x^2+1=0$, gives $x^2=-1$
$\Rightarrow x= \pm i$
$\therefore$ $x$ is not a real but $x$ is real(given)
$\therefore$ value of $x$ is possible.
Example 5: If $A$ is universal set, then $((A′)′)′ $ is empty set. (True/False)
Solution: Given that A is universal set.
$(A')'= A$
$((A')')' = A'$
$A' = A - A = \phi$
So, the statement is true.
Practice Questions
Test your understanding of set concepts with these quick and concept-based MCQs. These questions are designed to help you identify and apply key ideas like equal and equivalent set definition, notation, subsets, finite and infinite sets, along with other topics, through simple and effective practice.
Recommended Video based on Empty Sets
Watch this video for a quick and easy explanation of the empty set, its symbol, examples, and key properties along with equal and equivalent sets. It’s a helpful resource to understand the concept clearly in less time.
NCERT Useful Resources
Explore curated NCERT resources including notes, solutions, and exemplar problems covering all key set theory concepts like union of sets, intersection, power set, singleton set, subsets, and more. These materials help in understanding and support effective exam preparation.
NCERT Maths Class 11 Chapter 1 Sets Solutions
Frequently Asked Questions (FAQs)
Yes it is empty set or null set.
The other name of the empty set is the null set.
$\{0\}$ is not an empty set as it contains the element $0$ (zero).
Empty set doesn't have any element to count. However, the cardinality of an empty set is $0$.
A set that does not contain any element is empty set.
Example: $\mathrm{A}=$ $\{x: 9<x<10, x$ is natural number $\}$
An empty set is a set that contains no elements. It's a fundamental concept in set theory, representing a collection with nothing inside. The empty set is unique and is denoted by {} or ∅.
No, there is only one empty set. All empty sets are considered identical because they all contain the same thing: nothing. This is a key property of the empty set.
Yes, the empty set is a subset of every set, including itself. This is because every element in the empty set (of which there are none) is also an element of any other set.
The cardinality of an empty set is 0. Cardinality refers to the number of elements in a set, and since an empty set has no elements, its cardinality is zero.
Yes, an empty set can be an element of another set. For example, the set {∅} is not empty; it contains one element, which is the empty set.
The concept of a limit to the empty set is not well-defined in standard analysis. Limits are typically defined for sequences of numbers or points in a topological space, not for sets.
No, the empty set is not equal to zero. Zero is a number, while the empty set is a set. They are different mathematical concepts, although both represent a form of "nothingness" in their respective contexts.
In set-builder notation, the empty set can be represented as {x : x ≠ x}. This reads as "the set of all x such that x is not equal to x," which is impossible and thus results in an empty set.
There is no difference between a null set and an empty set. These terms are synonymous and both refer to a set that contains no elements.
The empty set has exactly one subset: itself. This is because the definition of a subset allows for the possibility of an "improper" subset, which is identical to the original set.
The power set of an empty set is {∅}. This is because the power set is the set of all subsets, and the empty set has only one subset (itself).
No, it's not possible to remove elements from an empty set because it doesn't contain any elements to begin with. Any operation to remove elements from an empty set will still result in an empty set.
Yes, you can perform a union operation with an empty set. The union of any set A with the empty set ∅ is always equal to A. Mathematically, A ∪ ∅ = A.
The intersection of any set with the empty set always results in the empty set. Mathematically, for any set A, A ∩ ∅ = ∅.
The empty set is considered finite. In set theory, a set is finite if it has a specific number of elements, and the empty set has exactly zero elements.
Yes, an empty set can be a proper subset of any non-empty set. A proper subset is a subset that is not equal to the original set, and the empty set fulfills this condition for all non-empty sets.
The complement of an empty set in a universal set U is the universal set itself. Mathematically, if ∅ represents the empty set, then ∅' = U.
No, you cannot have an empty set of empty sets. A set containing only the empty set, {∅}, is not itself empty; it has one element (the empty set).
The Cartesian product of an empty set with any other set always results in an empty set. This is because there are no elements in the empty set to pair with elements from the other set.
The concepts of "even" and "odd" don't apply to sets, including the empty set. These terms are used for integers, not sets.
Yes, in topology, the empty set is considered both open and closed simultaneously. This makes it one of the two sets (along with the entire space) that are always both open and closed in any topological space.
The result of taking the difference between a set and itself is always the empty set. For any set A, A - A = ∅.
Yes, you can define a function with an empty set as its domain. This is called an empty function. It's a valid function, but it doesn't map any elements because there are no elements in its domain.
When you take the symmetric difference of any set A with the empty set, the result is A itself. Mathematically, A ⊕ ∅ = A.
The empty set is considered countable. In set theory, finite sets are always countable, and the empty set is finite (with zero elements).
While you can write the symbol for an empty set infinitely many times, there is still only one unique empty set in mathematics. So, conceptually, you cannot have an infinite number of distinct empty sets.
Multiplication is not defined for sets, including the empty set. You can multiply the cardinality of sets, but since the cardinality of the empty set is 0, multiplying any number by 0 gives 0.
Yes, an empty set can be a solution to an equation, particularly when the equation has no solutions. For example, the solution set to the equation x^2 + 1 = 0 in the real number system is the empty set.
An empty set and a singleton set are different. An empty set has no elements, while a singleton set has exactly one element. However, a singleton set can contain an empty set as its single element, like {∅}.
Yes, you can perform set operations between two empty sets, but the result will always be an empty set. For example, ∅ ∪ ∅ = ∅, ∅ ∩ ∅ = ∅, and ∅ - ∅ = ∅.
Yes, the empty set is a subset of itself. This follows from the definition of a subset: every element in ∅ (of which there are none) is also an element of ∅.
The result of taking the union of all subsets of an empty set is the empty set itself. This is because the only subset of an empty set is the empty set.
No, an empty set cannot be considered as a partition of another set. A partition must cover the entire set, and an empty set covers nothing.
The dimension of an empty set in vector spaces is -1. This is a convention in linear algebra to maintain consistency with certain theorems.
Yes, you can define a metric on an empty set. Any function from ∅ × ∅ to the real numbers satisfies the metric axioms vacuously, so there is exactly one metric on the empty set.
No, an empty set cannot be dense in another set. For a set to be dense, it must have elements arbitrarily close to every point in the space, which an empty set cannot do.
The convex hull of an empty set is the empty set itself. This is because there are no points to form any convex combination.
Yes, the empty set can be considered a vector space, but only over the field with one element (often denoted as F1). This is a rather abstract concept in advanced algebra.
The closure of an empty set in any topological space is the empty set itself. This is because the empty set contains no points and has no limit points.
Yes, you can define a total order on an empty set. In fact, there is exactly one total order on the empty set, and it satisfies all the axioms of a total order vacuously.
Also Read
21 Jul'25 11:37 AM
02 Jul'25 07:43 PM
02 Jul'25 07:42 PM
02 Jul'25 06:39 PM
02 Jul'25 06:39 PM
02 Jul'25 06:38 PM
02 Jul'25 06:38 PM
02 Jul'25 06:38 PM
02 Jul'25 06:38 PM
02 Jul'25 06:38 PM