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.
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.\
Table of Content
- What is Disjoint Set?
- Disjoint Set Definition
- Condition for Disjoint Sets
- How to Check if Sets are Disjoint or Not?
- Disjoint Set Example
- Disjoint Set Venn Diagram
- Pairwise Disjoint Set
- Disjoint Union of Set
- Are Two Empty Sets Disjoint?
- Difference Between Joint and Disjoint Sets
- Solved Examples
- FAQs