Disjoint Sets

A well-defined group of unique items or components is called a set. Every item in a set is referred to as an element or set member.

Are Two Empty Sets Disjoint?

The definition and condition of disjoint sets state that sets are considered to be disjoint if their intersection gives an empty set. The condition for disjoint sets is satisfied when two empty sets intersect to give an empty set.
∅ ∩ ∅=∅
 

To determine if the given sets are disjoint, follow these steps:
Step 1: List the elements of each set.
Step 2: Check for common elements between the two sets.
Step 3: Apply the disjoint set condition, which is A ∩ B = ∅.
Step 4: The sets are disjoint if the condition is met; otherwise, they are not.
For example, verify if the sets A = { 30, 35, 49} and B = {6, 15} are disjoint.
Given sets,
A = {30, 35, 49} and B = {6, 15}
Check the condition: A ∩ B = ∅
A ∩ B = {30, 35, 49} ∩ {6, 15}
⇒ A ∩ B = ∅
Sets A and B are disjoint sets.

Properties of Disjoint Sets

The empty set (∅) is always the intersection of disjoint sets, since they have no elements in common.
An element that is a member of one disjoint set cannot be a member of the other, since they are mutually exclusive.
Disjoint sets in a Venn diagram are represented by non-overlapping circles, which makes it obvious that there is a shared area or element.

Disjoint Set Venn Diagram

Venn diagrams are used in set theory to show the sets. Since there is no common element in the sets, there are no common values in the Venn diagram for the disjoint sets A and B. The Venn diagram for disjoint sets A and B is as follows:

Pairwise Disjoint Set

A pairwise disjoint set is a collection of subsets. Suppose that A is a collection of sets. Let X and Y be the two sets in A. X and Y are referred to as pairwise disjoint sets if they are subsets of A, X ≠ Y, and X ∩ Y = ∅. Another name for the pairwise disjoint set is a mutually disjoint set. The mathematical definition of a pairwise disjoint set is:
X ⊆ A, Y ⊆ A, X ≠ Y and X ∩ Y=∅

Disjoint Union of Set 

The set operation that creates a set with all the elements of two sets is called the union. The regular union of sets combines all the elements from both sets, including any shared ones. On the other hand, the disjoint union refers to combining the sets that have no elements in common. A binary operation on two disjoint sets is the disjoint union. Following the disjoint union operation, the resulting set ought to meet the disjoint set condition. Another name for a set’s disjoint union is a discriminated union.
The ordered pair of elements that is (p, q), where q defines the index from which the element p is selected. That is present in the resultant set following the disjoint union of the set. We must make certain adjustments to the provided sets in order to perform the disjoint union of sets, and these adjustments and operations are as follows:
A ∪* B=(A × {0}) ∪ (B × {1})=A* ∪ B*
Where,
A and B are disjoint sets, ∪* represents the disjoint union.
 

Disjoint sets are used to enroll students in different classes, and more. Let us see how disjoint sets help in real life.

• Sports team selection
  There are three teams such as set A, B, and C. At a school, the football team (A), the volleyball team (B), and the basketball team (C). Since no students can play for more than one team, A, B, and C are disjoint sets.
• Categories of flight passengers
  Passengers are divided into three classes by the airline such as A, B, and C. The first class is set (A), business class as set (B), and economy class as set (C). These sets are disjoint because a passenger is only allowed to carry one class ticket per journey.
• Library book genres
  Books are arranged in a library according to three categories such as A, B, and C. The fiction as set (A), non-fiction as set (B), and comics as set (C). The sets A, B, and C are disjoint if every book is classified under a single genre; no book is a member of more than one genre.
• Blood donation eligibility
  Blood banks categorize donors into three groups, that is A, B, and C. The eligible donors are set (A), medically unfit donors are set (B), and donors who undergo testing are set (C). These sets are disjoint; there is no common element between them, because a person can only belong to one category.
• Employee departments
  The employees are assigned to set A, B, and C, such as HR as (A), finance as (B), and technology as (C) departments within a company. These sets are disjoint and have no common members. No employees work in more than one department concurrently.