Empty Set

Consider a set of all natural numbers less than $0$. This set doesn't have any elements as it is not possible to have a natural number less than $0$. So, such sets are called empty sets.The empty set is also known as the “null set”. Empty set is a type of set with no elements in it. Let's dive into the article to know more about empty sets.

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This Story also Contains

1. Set
2. Empty Set
3. Venn Diagram of Empty Set
4. Is Empty Set Finite or Infinite?
5. Difference between Zero Set and Empty Set
6. Empty set examples

This article will cover the concept of Empty Set. This concept falls under the broader category of sets relation and function, a crucial chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Set

It is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

Empty Set

A set that does not contain any element in it is called the empty set (or null set or void set).

Eg. $A=\{x: 1<x<2, x$ is a natural number $\}$
Since no natural number lies between 1 and 2, hence A will be an empty set.

The empty set is denoted by the symbol $\varphi$ or \{\}. Note: $\varphi \neq\{\varphi\}, \varphi \neq\{0\}$

What is Empty Set?

Consider the set
$\mathrm{A}=\{\mathrm{X}: \mathrm{x}$ is a student of Class XI presently studying in a school $\}$
We can go to the school and count the number of students presently studying in Class XI in the school. Thus, the set A contains a finite number of elements.

We now write another set $B$ as follows:
$B=\{x: x$ is a student presently studying in both Classes $X$ and $X I\}$
We observe that a student cannot study simultaneously in both Classes X and XI .
Thus, the set B contains no element at all.

According to this definition, $B$ is an empty set while $A$ is not an empty set. So, set B is the example of empty set. The empty set is denoted by the symbol $\varphi$ or $\{\}$

Empty Set symbol

The empty set is represented by the symbol “Φ” and “φ” or { }.(Read as 'phi')

Venn Diagram of Empty Set

They are the most effective tool to represent the relationships between sets, especially finite sets. Empty set can also be represented by venn diagrams.

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, X ∩ Y = ∅.

Properties of Empty Set

The properties of the empty set include:

Empty set symbol: The empty set is denoted by $\varphi$ or $\{\}$.

Cardinality: The number of elements of the empty set or the cardinality of empty set is zero, $|φ| = 0$

Empty set is a subset of every set: For any set A, the empty set is a subset of A, i.e. $φ ⊆ A; ∀ A$

Empty set subset: The only subset of an empty set is the empty set itself, i.e. $A ⊆ φ ⇒ A = φ$

Cartesian product of empty set: The Cartesian product of A and the empty set is the empty set, i.e. $A × φ = φ; ∀ A$

Power set of empty set: The power set of the empty set is the set containing only the empty set, i.e. $2φ = {φ}$

Union with empty set: The union of A with the empty set is the set A itself, i.e. $A ⋃ φ = A; ∀ A$

Intersection with empty set :The intersection of A with the empty set is again the empty set, i.e. $A ⋂ φ = φ; ∀ A$

Remarks:-
1. $\phi$ is called the null set.
2. $\phi$ is unique.
3. $\phi$ is a subset of every set.
4. $\phi$ is never written within braces i.e., $\{\phi\}$ is not the null set.
5. $\{0\}$ is not an empty set as it contains the element $0$ (zero).

$
\text { Ex- }\{x: x \in N, 4<x<5\}=\phi
$

Is Empty Set Finite or Infinite?

An empty set is a finite set since its cardinality is defined and is equal to $0$. As we know, a set is said to be infinite if the number of elements in it are infinite, i.e. its cardinality is $∞$ or not defined, whereas a finite set contains a countable number of elements.

Difference between Zero Set and Empty Set

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

As we learned

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
As we learn
EMPTY SET-
A set which does not contain any element is called the empty set or the null set or the void set.
- wherein
eg. $\{1<x<2, x$ 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.

$(A')'= A$

$((A')')' = A'$

$A' = A - A = φ$

So, the statement is true.

List of Topics Related to Empty Sets