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Study About Set Operations on Empty Set Here

Before learning about the operations that can be performed on an empty set, let’s first understand what empty sets are.

Empty Set (Null Set)

An Empty Set, also known as a null set, has no elements. The sign ‘∅’ denotes an empty set. It is spelled ‘phi’. For example, Set X ={}. It is sometimes referred to as a void set or a null set.

For example, a month with 35 days, a week with two Sundays, or a horse with eight legs. 

Here are some additional examples of how to classify odd elements using an empty set.

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

The prime number set will be referred to as A. Thus, A = {2, 3, 5, 7, 11, 13, 17…}. Because there are no prime numbers between 14 and 16, we can conclude that the X is an empty set.

  • The number of vans having 10 doors.

In actuality, getting a vehicle with 10 doors is impossible unless a van manufacturing firm designs a specific model. As a result, the set with the ten-door van is empty.

To have a better understanding, let’s go over the operations on an Empty Set one by one using examples.

Properties of Empty Sets

Given that there is just one empty set, it’s worth investigating what occurs when the set operations intersection, union, and complement are applied to the empty set and a general set indicated by X. Let’s go over each one in depth.

Union with an Empty Set

A set operation between any set and an empty set always returns the set. XU∅ = X denotes the union of any finite or infinite set X with an empty set. Because an empty set has no components of its own, its union with any set X produces the identical set X.

For example,

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

The union of a given set X and an empty set is expressed as XU∅ ={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 always returns the set. X ∩ ∅ = X denotes the union of any finite or infinite set X with an empty set. Because an empty set has no elements of its own, any non-empty set and an empty set will share no elements.

For example,

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

The union of a given set X and an empty set is expressed as X ∩ ∅ = {2, 4, 6}.

Difference between a Set and Empty Set 

The only difference between a set X and an empty set is set X itself:

X-∅=X

Complement of Empty Set

The complement of the empty set is the universal set for the setting in which we are operating. This is because the set of all elements that aren’t in the empty set is the same as the set of all elements.

Cartesian Product of Empty Set

The Cartesian product of a set and an empty set, such as set A and empty set X × φ = φ, ∀ X. This also implies that the cartesian product of a set and 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

Important Notes on Empty Set

Here is a list of some key points about an empty set.

  • If a set has no elements, it is defined as an empty set or a null set.
  • Empty sets are also referred to as void sets or null sets.
  • The union operation on any set and an empty set always yields the set itself.
  • Since the cardinality of an empty set is defined and equal to zero, it is a finite set.

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