Empty Set

An empty set, called a null or void set, is a set that contains no elements. 

Here are a few examples of empty sets. 

A = {x | x is a prime number and 11 < x < 13}

B = {x | x is prime number and x < 2}

C = {x | x³ = 8 and x is an odd number}

Symbol

It is represented by the symbol ‘ɸ’ (phi) and the notation ɸ = { }

Venn Diagram

Let A = {m, n, p, q} and set B = {b, c, d, e}

Since A ∩ B = ɸ is an empty set, on representing it in a Venn diagram, 

Here, set A = {m, n, p, q} and set B = {b, c, d, e}

The intersection of sets A and B is an empty set. Thus, A ∩ B = ɸ

Properties

Cardinality

Since an empty set has no elements, its cardinality is zero. 

|ɸ| = 0, where ‘ɸ’ is an empty set.

Since the cardinality of an empty set is 0, it is considered a finite set.

Subset 

An empty set has no subset other than itself.

If A ⊆ ɸ, then A = ɸ

Subset of Every Set

An empty set is always the subset of a given set.

For any set ‘A,’ the empty set is a subset of the set ‘A.’ That is, ɸ ⊆ A, ∀ A

For example,

If A = {7, 21, 35}, its subsets are ɸ, {7}, {21}, {35}, {7, 21}, {21, 35}, {7, 35}, {7, 21, 35}. 

Thus ɸ ⊆ A

Union 

Since an empty set contains no elements, its union with any set gives the same set.

If ‘A’ is a set and ‘ɸ’ is an empty set, then the union of A and ɸ is A ∪ ɸ = A

For example,

If A = {7, 21, 35} and ɸ are two sets, then A ∪ ɸ = {7, 21, 35} ∪ ɸ = {7, 21, 35}

Intersection 

The intersection of any set with the empty set is the empty set itself.

If ‘A’ is a set and ‘ɸ’ is an empty set, then the intersection of A and ɸ is A ∩ ɸ = ɸ

For example,

If A = {7, 21, 35} and ɸ are two sets, then A ∩ ɸ = {7, 21, 35} ∩ ɸ = ɸ

Power Set 

The power set of an empty set contains elements only of the empty set. 

P(ɸ) = { } = ɸ, where ‘ɸ’ is an empty set.

Cartesian Product 

The cartesian product of any set and the empty set gives the empty set.

If ‘A’ is a set and ‘ɸ’ is an empty set, then the cartesian product A × ɸ = ɸ, ∀ A

If A = {7, 21, 35} and ɸ is the empty set, then A × ɸ = {7, 21, 35} × ɸ = ɸ

Complement 

The complement of the empty set gives the universal set. 

If A = ɸ, then A’ = (ɸ)’ = U

Here, ‘U’ is the universal set.

Zero Set vs. Empty Set

Zero Set	Empty Set
It contains only zero as its element.	It contains no element.
It is represented as {0}	It is represented as { }
The cardinality is 1	The cardinality is 0
For example, A = {x | x < 1 and x is a whole number}B = {0}C = {Set of whole numbers between -1 and 1}	For example, A = {Set of natural numbers less than 0}B = { }C = {x | x is a composite number and 2 < x < 3}

Solved Examples