Empty Set

In Mathematics, two of the most basic topics are equality and equivalence of sets having instances that help in analyzing the relation between two or more sets or empty sets. For example, let us find two teachers who have planned for a school trip out. Mrs. Johnson fills a cooler with lunch, each container in the cooler has a lunch box symbolizing every student in her class, ten in total. At the same time, another teacher who also participated is Mr. Lee, who educates his ten learners. If we represent each cooler as a set of lunch boxes then we can say that the sets are equal in terms of the size, which is ten lunch boxes.

This article will cover the concept of Empty Sets. 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:

A set 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…...

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

Properties of Empty Set:

A set having no element is called a null set empty set or void set. it is denoted by $\varphi$ or \{\} .

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
$
Summary: The empty set is a fundamental concept in mathematics. Despite having no elements, it is considered a subset of every other set. The concept of empty set is used in various fields of mathematics and real life.

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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

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 an 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 the 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 $x^2=-1$

$\Rightarrow x= \pm i$

x is not a real but x is real(given)

value of x is possible.

FAQs:
1) What is an empty set:

Solution: A set that does not contain any element in it is called the empty set (or null set or void set).
2) What is the other name of the empty set?

Solution: The other name of the empty set is the null set.
3) Is zero the empty set?

Solution: $\{0\}$ is not an empty set as it contains the element 0 (zero).
4) Is Empty set countable?

Solution: Yes, the empty set is countable. However, the cardinality of an empty set is 0 .
5) What is a set?

Solution: What is a set?
Ans: A set is a collection of distinct objects, considered whole. These objects are called elements or members of the set.