Empty Set: Class 11 Maths Notes, Definition, Representation & Practice Examples

The main empty set meaning is that it is a set with no elements. A set having no object or element defines a situation where there is no possible outcome.  

Importance of Empty Set for JEE Mains and other Engineering Exams

Some of the key areas where empty sets are applicable in maths are below. You will also see how JEE or other engineering exams ask candidates about empty sets across the same. 

Area	Role of Empty Set	JEE Mains Application Example
Set Theory	Acts as subset of every set	Counting subsets, Venn diagrams
Algebra	Solution set when no solution exists	Solving equations, inequalities
Probability	Represents impossible events (P = 0)	Probability of impossible events
Topology	Trivial open/closed set	Properties of intervals, unions, intersections

 

In the NCERT Class 11 textbook, in Chapter 1, the empty set is in section, 1.3. It states

 

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

According to this definition, B is an empty set while A is not an empty set. The empty set is denoted by the symbol φ or { }."

 

Basic explanation of Empty Set that you can easily remember. 

• An empty set has no elements, as nothing fits within the condition it describes.
• We use the empty set sign as φ or { } to quickly show that a set has nothing in it.

 

 

 

Now that you know what is an empty set, let's look at some simple real-world empty sets examples. 

1. A month with $32$ days

2. A week with two Sundays

3. An animal with five legs

4. A car with ten doors

In all these cases, we are looking for something impossible or nonexistent, which leads to an empty set.

Here we are going to show the empty set symbol. 
Empty $A=\left\{\right\}$ sets are usually represented in two main ways:

1. Curly Braces: $\left\{\right\}$

2. Special Symbol: $\varnothing$ (phi)

Example:

Let  ( A is an empty set)

Or A = $\varnothing$

Important: Do not confuse $\left\{\varnothing \right\}$ with $\mathrm{\varnothing }$ . $\left\{\varnothing \right\}$  is a set containing one element, the empty set itself.

Understanding with an Example

Set $A=\{y:y$  is a multiple of 4 and $5<y<\bar{7}\}$

Multiples of 4 are $\mathrm{4,8},12$ , etc. Between $5$  and $7$ , there are no multiples of $4$ .

Thus, Set $A$ has no elements: $A=\varnothing$

Subset Property:

An empty set is a subset of every set.

 

Union with Empty Set:

$A\cup \mathrm{\varnothing }=A$

 

Intersection with Empty Set:

$A\cap \mathrm{\varnothing }=\varnothing$

 

Cardinality of Empty Set:

$\left|\varnothing \right|=0$

 

Cartesian Product with Empty Set:

$A\times \mathrm{\varnothing }=\varnothing$

1. $\varnothing \ne \left\{\varnothing \right\}$

2. Power set of $\mathrm{\varnothing }$ is $\left\{\varnothing \right\}$

3. Number of subsets of $\mathrm{\varnothing }=1$

4. Empty set is both open and closed in Topology

5. Probability of an empty event $=0$

1. Find the number of elements in the power set of an empty set.

Solution: Power set has $2^{\mathrm{n}}$  elements. Here, $\mathrm{n}=0$ , thus $2^0=1$  element $\to \left\{\varnothing \right\}$ .

2. Find $A\times \mathrm{\varnothing }$ where $A=\{\mathrm{1,2},3\}$ .

Solution: $\mathrm{A}\times \varnothing =\varnothing$ .

3. Is $\mathrm{\varnothing }$ a subset of $\mathrm{\varnothing }$ ?

Solution: Yes. Every set is a subset of itself.