Set Operations

Sets are collections of unique objects or elements, and set operations are mathematical operations carried out on sets.

What is Set Operation

Set operations are mathematical operations performed operation on sets, which are collections of distinct objects or elements. These operations are fundamental in set theory and are used to manipulate sets, define relationships between sets, and solve problems involving collections of objects.

There are three major types of operations performed on sets, such as:

• Union of sets
• Intersection of sets
• Disjoint
• Set Difference 
• Complement
• Addition & Subtraction 

Let us discuss these operations one by one.

Union

Union of sets A and B, denoted by A ∪ B, is the set of distinct elements that belong to set A or set B, or both.

Venn diagram of A ∪ B

Above is the Venn Diagram of A U B.

Example: Find the union of A = {2, 3, 4} and B = {3, 4, 5}; 
Solution : A ∪ B = {2, 3, 4, 5}.

Intersection

The intersection of the sets A and B, denoted by A ∩ B, is the set of elements that belong to both A and B i.e. set of the common elements in A and B.

Venn diagram of A ∩ B

Above is the Venn Diagram of A ∩ B.

Example: Find the intersection of A = {2, 3, 4} and B = {3, 4, 5} 
Solution : A ∩ B = {3, 4}.

Disjoint

Two sets are said to be disjoint if their intersection is the empty set. i.e, sets have no common elements.

Above is the Venn Diagram of A disjoint B.

Example: Let A = {1, 3, 5, 7, 9} and B = { 2, 4, 6, 8}  
A and B are disjoint sets since both of them have no common elements.

Set Difference

The difference between sets is denoted by ‘A – B’, which is the set containing elements that are in A but not in B. i.e., all elements of A except the element of B.

Above is the Venn Diagram of A-B.

Example: If A = {1, 2, 3, 4, 5} and B = { 2, 4, 6, 8}, find A-B
Solution: A-B = {1, 3, 5}

Complement

The complement of a set A, denoted by A^C is the set of all the elements except the elements in A. Complement of the set A is U – A.

Above is the Venn Diagram of A^c

Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8}.
Find AC
Solution: AC = U-A = {1, 3, 5, 7, 9, 10}

Addition & Subtraction

Addition of sets A and B, referred to as Minkowski addition, is the set in whose elements are the sum of each possible pair of elements from the 2 sets (that is one element is from set A and the other is from set B).
Set subtraction follows the same rule, but with the subtraction operation on the elements. It is to be observed that these operations are operable only on numeric data types. Even if operated otherwise, it would only be a symbolic representation without any significance. Further, it can be seen easily that set addition is commutative, while subtraction is not. 

For addition and consequently subtraction, please refer to this answer.

[Tex]A-B=A\cap \bar{B}  [/Tex]

Check: Set Operations practice

Property of set operation

Set operations have several important properties that govern their behavior. Here are some fundamental properties of set operations:

Closure Property:

• Set operations are closed under their respective operations, meaning that performing an operation on sets results in another set.
• For example, the union, intersection, and difference of sets always produce sets as their results.

Commutative Property:

• Union: A ∪ B = B ∪ A
• Intersection: A ∩ B = B ∩ A
• Symmetric Difference: A Δ B = B Δ A

Associative Property:

• Union: (A ∪ B) ∪ C = A ∪ (B ∪ C)
• Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Property:

• Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Intersection over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Identity Property:

• Union: A ∪ ∅ = A
• Intersection: A ∩ U = A, where U represents the universal set
• Symmetric Difference: A Δ ∅ = A

Complement Property:

• Union: A ∪ A’ = U, where U is the universal set
• Intersection: A ∩ A’ = ∅ (the empty set)

Absorption Property:

• Union over Intersection: A ∪ (A ∩ B) = A
• Intersection over Union: A ∩ (A ∪ B) = A

Conclusion of Set Operation

Set operations are essential tools that let us work with, combine, and evaluate groups of items or components. We may represent real-world settings, describe relationships between sets, and solve set-related problems by using operations like union, intersection, complement, difference, and symmetric difference. A systematic foundation for comprehending and thinking about sets is provided by the qualities of set operations, which include closure, commutative, associative, distributive, identity, complement, and absorption features.

Set Operations – FAQs

What are the 4 operations of sets?

The four fundamental operations of sets, commonly used in set theory and mathematics, are:

• Union
• Intersection
• Difference
• Complement

What are operations 4?

• They are addition, subtraction, multiplication, and division.

What is the union symbol?

“∪” serves as the union symbol, which symbolizes the process of joining two sets to create a new set that includes every unique element from both sets.

What is power of a set?

The Set, including the empty set and the set itself, is called the power set of that set. the power set of a set with n elements will have 2 n subsets if the set has n elements.

What is the formula for set operation?

n(P∪Q) = n(P) + n(Q) − n(P∩Q).