Empty Set

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 the empty set is either $\emptyset$ or $\{\}$. Is $Ø$ an empty set? Yes ,it is an 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:

1. 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, $A = \emptyset$.
2. 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, $B = \emptyset$.
3. 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:

Remarks on Empty Set

1. $\phi$ is called the null set.
2. $\phi$ is unique, meaning there is only one empty set.
3. $\phi$ is a subset of every set.
4. $\phi$ is not written within braces; that is, ${\phi}$ is not the empty set.
5. ${0}$ 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 the Empty Set Finite or Infinite?

Is the Empty set countable? An empty set doesn't have any elements 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 a Zero Set and an Empty Set?

Though they may sound similar, the zero set and the empty set are different. This section explains how they differ based on elements, notation, and meaning in set theory.

Why is the Empty Set a Subset of Every Set?

One of the most important properties of the empty set in mathematics is that it is considered a subset of every set.

At first, this might feel strange. You might wonder, “How can a set with no elements be inside every set?”

But think about it calmly.

A subset simply means that all elements of one set must belong to another set. Now, the empty set (null set) has no elements at all, so there are no elements that can violate this rule. Since nothing contradicts the subset condition, the empty set automatically satisfies it for every set.

In simple words: If there is nothing inside the empty set, there is nothing that can be outside another set.

So, it safely fits inside every set.

Because of this, in set theory, we always write:

$\emptyset \subseteq A$ for any set $A$.

This is a standard rule and is widely used in subset problems, proofs, and mathematical logic.

Proof Using Subset Definition

Let’s now understand this formally using the definition of a subset.

By definition: Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$.

Mathematically, $A \subseteq B \iff \forall x (x \in A \Rightarrow x \in B)$

Now take $A = \emptyset$.

Since the empty set has no elements, there is no element $x$ in $\emptyset$ to check.

So the condition “for every element in $A$” becomes automatically true, because there are no elements to test.

This type of statement is called vacuously true in mathematics.

Therefore, $\emptyset \subseteq B$

for every possible set $B$.

Example

Suppose $A = \{1,2,3\}$

Now ask: “Is there any element in $\emptyset$ that is not in $A$?”

No - because $\emptyset$ has nothing.

So we conclude: $\emptyset \subseteq A$

Empty Set Solved 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 option 3.

Example 2: Which of the following sets is the 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 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 option 4.

Example 4: Which of the following is an 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 isthe universal set.
$(A')'= A$
$((A')')' = A'$
$A' = A - A = \phi$
So, the statement is true.

List of related topics to Empty Set

To build a strong understanding of the Empty Set, it's helpful to explore other key set theory concepts that often relate to it. Topics such as Equal and Equivalent Sets, Subsets, Finite and Infinite Sets, Singleton Set, Power Set, and Universal Set provide essential context and connections. Learning these concepts together will enhance your ability to classify and analyse sets with precision.

NCERT Useful Resources

Explore curated NCERT resources including notes, solutions, and exemplar problems covering all key set theory concepts like union of sets, intersection of sets, 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

NCERT Maths Class 11 Chapter 1 Sets Notes

NCERT Maths Exemplar Problems Class 11 Chapter 1 Sets

Practice Questions based on Sets

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.