Let’s first understand what is an empty set before learning about the operations that can be performed on it.

Empty Set (Null Set)

An Empty Set, also known as a null set, is one that has no elements. The symbol ‘∅’ represents an empty set. It is pronounced ‘phi.’ For eg, Set X = {} . It is also known as a void set or a null set.

For instance, a month with 33 days, a week with two Tuesdays, or a cat with five legs. 

Here are some more examples of how an empty set can be used to classify odd elements.

• X = {x : x is a prime number and 14<x<16}

We’ll refer to the prime number set as A. Thus, A = {2, 3, 5, 7, 11, 13, 17…}. We can deduce that the X is an empty set because there are no prime numbers between 14 and 16.

• The number of vans with 12 doors.

In reality, unless a van manufacturing company builds a specific model, finding a van with 12 doors is impossible. As a result, the set featuring the van with 12 doors is empty.

Let’s now go ahead and learn about operations on an Empty Set one by one with examples to get better clarity.

Properties of Empty Sets

Because there is only one empty set, it is interesting to see what happens when the set operations intersection, union, and complement are applied to the empty set and a general set denoted by X. Let’s discuss each of them in detail.

Union with an Empty Set

A set operation between any set and an empty set will always result in the set itself. The union of any finite or infinite set X with an empty set is given by X U ∅ = X. Because an empty set has no components of its own, the union of an empty set with any set X yields the same set X.

For example,

Consider a set X = {1, 2, 3, 4}.

The union of the given set X with an empty set can be written as X U ∅ = {1, 2, 3, 4} U { }. Thus, A U ∅ = {1, 2, 3, 4}

Intersection with an Empty Set

The intersection of sets operation between any set and an empty set will always result in the set itself. The union of any finite or infinite set X with an empty set is given by X ∩ ∅ = X. Because an empty set has no elements of its own, there will be no element in common between any non-empty set and an empty set.

For example,

Consider a set X = {2, 4, 6}.

The union of the given set X with an empty set can be written as X ∩ ∅ = {2, 4, 6}

Difference between a Set and Empty Set 

The difference between a set X and the empty set is set X itself:

X-∅=X

Complement of Empty Set

The universal set for the setting in which we are working is the complement of the empty set. Because the set of all elements that aren’t in the empty set is the same as the set of all elements, this is the case.

Cartesian Product of Empty Set

The Cartesian product of a set and an empty set, for example, set A and an empty set X × φ = φ, ∀ X. This further means that a set’s cartesian product with an empty set is always an empty set.

Assume a non empty set X = {1, 2, 3, 4} and an empty set = {}.

Their cartesian product =X × φ = φ