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)}
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
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