Disjoint Sets

Disjoint Sets are one of the types of many pair of sets, which are used in Set Theory, other than this other types are equivalent sets, equal sets, etc. Set Theory is the branch of mathematics that deals with the collection of objects and generalized various properties for these collections of objects.

In this article, we will learn about Disjoint Sets in detail including their definition, condition, and Venn diagram. We will also learn about how to check disjoint sets and disjointed unions of sets along with the examples. Disjoint sets are used in various fields of mathematics and data structures. Let’s start our learning on the topic of Disjoint Sets.

What is Disjoint Set?

Disjoint sets are also known as disjoint collections or non-intersecting sets and refer to those sets which don’t have any identical element common between them. For example, A = {1, 2, a} and B = {3, 4, b} then, A ∩ B = ϕ. For this example, A and B are disjoint sets as both do not contain the same element.

Disjoint Set Definition

Disjoint sets are sets that do not contain any common element. The sets which when intersected results in an empty set or null set are called Disjoint Sets. The disjoint sets are also called mutually exclusive sets or independent sets.

If P = {a, b} and Q = {c, d} then, P ∩ Q = ϕ as there is no same element in set P and set Q. Thus, P and Q are called disjoint sets.

Condition for Disjoint Sets

If A and B are two disjoint sets then, their intersection must be an empty set i.e.,

A ∩ B = ϕ

How to Check if Sets are Disjoint or Not?

To check whether the given sets are disjoint or not we follow below steps:

Step 1: Compare the elements of given sets.

Step 2: Compute the intersection of these sets.

Step 3: Apply the disjoint set condition i.e., A ∩ B = ϕ.

Step 4: If condition satisfies, the sets are disjoint otherwise not.

Disjoint Set Example

Example: Check whether the given sets A = {20, 25, 39} and B = {5, 19} are disjoint or not.

Answer:

Given sets,

A = {20, 25, 39} and B = {5, 19}

Check the condition: A ∩ B = ϕ

A ∩ B = {20, 25, 39} ∩ {5, 19}

⇒ A ∩ B = ϕ (condition satisfied)

Sets A and B are disjoint sets.

Disjoint Set Venn Diagram

In set theory, we represent the sets in the form of Venn diagrams. In the Venn diagram for disjoint sets, there is no overlapping of sets as there is no common element present in the sets.

Following is the Venn diagram for disjoint sets A and B:

Learn more about Venn Diagram.

Pairwise Disjoint Set

Pairwise Disjoint set is the group of subsets. If A is the set of collection of sets. X and Y be the two sets in A. If X and Y are subsets of A, X ≠ Y, and X ∩ Y = ϕ, then X and Y are called pairwise disjoint sets. The pairwise disjoint set is also called a mutually disjoint set. The mathematically pairwise disjoint set is given as:

X ⊆ A, Y ⊆ A, X ≠ Y and X ∩ Y = ϕ

Disjoint Union of Set

Union is the set operation performed on two sets resulting in a set containing all the elements of two sets. The disjoint union of sets is different from the regular union operation of sets. The disjoint union is a binary operation on two disjoint sets. After performing the disjoint union operation, the resultant set should satisfy the condition for the disjoint set. The disjoint union of a set is also called discriminated union as well.

The resultant set after the disjoint union of the set contains the element in ordered pair (p, q) where q defines the index from where element p is selected. For performing the disjoint union of sets, we have to do some changes in the given sets, and these changes and operation is given as follows:

A ∪* B = (A × {0}) ∪ (B × {1}) = A* ∪ B*

Where,

• A and B are disjoint sets,
• ∪* represents the disjoint union.

Example: If P = {9, 12} and Q = {5, 13} are disjoint sets then find a disjoint union of sets.

Answer:

P ∪* Q = (P × {0}) ∪ (Q × {1}) = P* ∪ Q*

P* = (P × {0})

⇒ P* = {(9, 0), (12, 0)}

and Q* = (Q × {1})

⇒ Q* = {(5, 1), (13, 1)}

P ∪* Q = P* ∪ Q*

⇒ P ∪* Q = {(9, 0), (12, 0)} ∪ {(5, 1), (13, 1)}

⇒ P ∪* Q = {(9, 0), (12, 0), (5, 1), (13, 1)}

Are Two Empty Sets Disjoint?

According to the definition and condition of the disjoint sets, if the intersection of sets results in an empty set, then the sets are called disjoint sets. When we perform the intersection of two empty sets it results in an empty set i.e., it satisfies the condition for disjoint sets.

ϕ ∩ ϕ = ϕ

Thus, the empty set is a disjoint set of itself.

Difference Between Joint and Disjoint Sets

Characteristics	Joint Set	Disjoint Set
Definition	The sets which have common elements i.e.,  the intersection of the set is not an empty set are  called joint sets.	The sets which do not have common elements i.e.,  the intersection of the set is an empty set is called  disjoint sets.
Condition	A ∩ B ≠ ϕ	A ∩ B = ϕ
Example	If X = {a, b} and Y = {b, c} then,  X ∩ Y = {b} X and Y are joint sets.	If X = {a, d} and Y = {b, c} then,  X ∩ Y = ϕ X and Y are disjoint sets.

Read, More

• Union of Sets
• Intersection of Sets
• De Morgan’s Law

Solved Examples on Disjoint Sets

Example 1: Check whether the given sets A = {p, q} and B = {r, s} are disjoint or not.

Answer:

Given sets,

A = {p, q} and B = {r, s}

Check the condition: A ∩ B = ϕ

⇒ A ∩ B = {p, q} ∩ {r, s}

⇒ A ∩ B = ϕ (condition satisfied)

Sets A and B are disjoint sets.

Example 2: Show that the sets P = {20, 40} and Q = {10, 30, 50} are disjoint sets.

Answer:

Given sets,

P = {20, 40} and Q = {10, 30, 50}

Check the condition: P ∩ Q = ϕ

⇒ P ∩ Q = {20, 40} ∩ {10, 30, 50}

⇒ P ∩ Q = ϕ (condition satisfied)

Sets P and Q are disjoint sets.

Example 3: Determine whether the given sets X = {12, 39, 48} and Y = {15, 60} are disjoint or not.

Answer:

Given sets,

X = {12, 39, 48} and Y = {15, 60}

Check the condition: X ∩ Y = ϕ

⇒ X ∩ Y = {12, 39, 48} ∩ {15, 60}

⇒ X ∩ Y = ϕ (condition satisfied)

Sets X and Y are disjoint sets.

Example 4: If P = {2, 7, 12} and Q = {8, 14} are disjoint sets then find a disjoint union of sets.

Answer:

P ∪* Q = (P × {0}) ∪ (Q × {1}) = P* ∪ Q*

P* = (P × {0})

⇒ P* = {(2, 0), (7, 0), (12, 0)}

and Q* = (Q × {1})

⇒ Q* = {(8, 1), (14, 1)}

P ∪* Q = P* ∪ Q*

⇒ P ∪* Q = {(2, 0), (7, 0), (12, 0)} ∪ {(8, 1), (14, 1)}

⇒ P ∪* Q = {(2, 0), (7, 0), (12, 0), (8, 1), (14, 1)}

Example 5: Which of the following sets are disjoint?

(i) A = {3, 18} and B = {9, 17}

(ii) X = {2, 4} and Y = {4, 8}

Answer:

(i) A = {3, 18} and B = {9, 17}

First, compute A ∩ B

⇒ A ∩ B = {3, 18} ∩ {9, 17}

⇒ A ∩ B = ϕ

Above sets satisfies the condition for disjoint sets.

A and B are disjoint sets.

(ii) X = {2, 4} and Y = {4, 8}

First, compute X ∩ Y

⇒ X ∩ Y= {2, 4} ∩ {4, 8}

⇒ X ∩ Y= {4}

⇒ X ∩ Y ≠ ϕ

Above sets does not satisfies the condition for disjoint sets.

X and Y are not disjoint sets.

FAQs on Disjoint Sets

1. Define Disjoint Sets.

Disjoint sets are the sets which do not contain any common element i.e., intersection of sets results in an empty set.

2. Write the Condition for Disjoint Sets.

If A and B are two disjoint sets then, their intersection must be an empty set.

A ∩ B = ϕ

3. How to find Disjoint Sets?

To find the disjoint sets first compute the intersection of given sets and check with the condition A ∩ B = ϕ, if the intersection is an empty set, then given sets are disjoint sets otherwise not.

4. Can Two Empty Aets be Disjointed?

When we apply intersection in two empty sets it results in empty set which satisfies the condition for the disjoint set. So, the empty sets can be disjointed, and empty set is disjoint set of itself.

5. What are Mutually Disjoint Sets?

Two sets are said to be mutually disjoint sets when it satisfies the following conditions:

X ⊆ A, Y ⊆ A, X ≠ Y and X ∩ Y = ϕ

6. Write the Set Representation for the Disjoint Union of a Set.

The set representation for disjoint union of set is given by:

A ∪* B = (A × {0}) ∪ (B × {1}) = A* ∪ B*

Where,

• A and B are Disjoint Sets
• ∪* represents the Disjoint Union