Empty Set

Empty Set (Null Set) - Definition and Examples

The empty set, also known as the null set, is one of the most basic and important concepts in set theory. It refers to a set that contains no elements. It is often the starting point for understanding how sets behave.

Definition of Empty Set

An empty set is a set that has no elements in it. It is represented by:

• $\mathrm{\varnothing }$ (the Greek letter phi)
• or $\{\}$ (a pair of curly brackets with nothing inside)

Notation

• Symbolically: $\mathrm{\varnothing }$ or $\{\}$
• Read as: “phi” or “the empty set”

Characteristics of the Empty Set

• It has zero elements.
• It is a subset of every set.
• The number of elements in the empty set is:

$n(\mathrm{\varnothing })=0$

• It is a finite set because it contains a countable number of elements (zero).

Examples of Empty Set

Example 1:

The set of even prime numbers greater than 2. $A=\{\text{even primes }>2\}$ This set is empty because 2 is the only even prime number.

Example 2:

The set of square roots of -9 in the set of real numbers. $B=\{\sqrt{-9}\in \mathbb{ℝ}\}$ No real number satisfies this condition, so the set is empty.

Example 3:

The set of students in Class 10 who are 2 years old. $C=\{\text{students of age 2 in Class 10}\}$ This is not possible, so the set is empty.

Is the Empty Set a Subset?

Yes, the empty set is a subset of every set. This means:

$\mathrm{\varnothing }\subseteq A$ for any set A

This is because there is no element in the empty set that violates the condition of being in another set.

Is There Only One Empty Set?

Yes, there is only one unique empty set. All empty sets are equal because they all contain exactly zero elements.

Importance of Empty Set

• It is the foundation for many set theory rules.
• Used to define concepts such as:

 * Disjoint sets
 * Power sets
 * Set complements

• Common in mathematics, logic, and computer science.

Conclusion

The empty set or null set is a set that has no elements. It plays a critical role in set theory and is a fundamental concept in both pure and applied mathematics. Understanding it helps students grasp more complex set operations and logical reasoning.