Set Operations

Set operations is a concept similar to fundamental operations on numbers. Sets in math deal with a finite collection of objects, be it numbers, alphabets, or any real-world objects. Sometimes a necessity arises wherein we need to establish the relationship between two or more sets. There comes the concept of set operations.

There are four main set operations which include set union, set intersection, set complement, and set difference. In this article, we will learn the various set operations, notations of representing sets, how to operate on sets, and their usage in real life.

A set is defined as a collection of objects. Each object inside a set is called an 'Element'. A set can be represented in three forms. They are statement form, roster form, and set builder notation. Set operations are the operations that are applied on two or more sets to develop a relationship between them. There are four main kinds of set operations which are as follows.

• Union of sets
• Intersection of sets
• Complement of a set
• Difference between sets/Relative Complement

Before we move on to discuss the various set operations, let us recall the concept of Venn diagrams as it is important in understanding the operations on sets. A Venn diagram is a logical diagram that shows the possible relationship between different finite sets. The Venn diagram can be represented as follows.

Now that we know the concept of a sets and Venn diagram, let us discuss each set operation one by one in detail. The various set operations are:

Union of Sets

For two given sets A and B, A∪B (read as A union B) is the set of distinct elements that belong to set A and set B or both. The number of elements in A ∪ B is given by n(A∪B) = n(A) + n(B) − n(A∩B), where n(X) is the number of elements in set X. To understand this set operation of the union of sets better, let us consider an example: If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then the union of A and B is given by A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

Intersection of Sets

For two given sets A and B, A∩B (read as A intersection B) is the set of common elements that belong to set A and B. The number of elements in A∩B is given by n(A∩B) = n(A)+n(B)−n(A∪B), where n(X) is the number of elements in set X. To understand this set operation of the intersection of sets better, let us consider an example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the intersection of A and B is given by A ∩ B = {3, 4}.

Set Difference

The set operation difference between sets implies subtracting the elements from a set which is similar to the concept of the difference between numbers. The difference between sets A and set B denoted as A − B lists all the elements that are in set A but not in set B. To understand this set operation of set difference better, let us consider an example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the difference between sets A and B is given by A - B = {1, 2}.

Complement of Sets

The complement of a set A denoted as A′ or A^c (read as A complement) is defined as the set of all the elements in the given universal set(U) that are not present in set A. To understand this set operation of complement of sets better, let us consider an example: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then the complement of set A is given by A' = {5, 6, 7, 8, 9}.

The above image shows various set operations with the help of Venn diagrams. When the elements of one set B completely lie in the other set A, then B is said to be a proper subset of A. When two sets have no elements in common, then they are said to be disjoint sets. Now, let us explore the properties of the set operations.

The properties of set operations are similar to the properties of fundamental operations on numbers. The important properties on set operations are stated below:

• Commutative Law - For any two given sets A and B, the commutative property is defined as,
  A ∪ B = B ∪ A
  This means that the set operation of union of two sets is commutative.
  A ∩ B = B ∩ A
  This means that the set operation of intersection of two sets is commutative.
• Associative Law - For any three given sets A, B and C the associative property is defined as,
  (A ∪ B) ∪ C = A ∪ (B ∪ C)
  This means the set operation of union of sets is associative.
  (A ∩ B) ∩ C = A ∩ (B ∩ C)
  This means the set operation of intersection of sets is associative.
• De-Morgan's Law - The De Morgan's law states that for any two sets A and B, we have (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
• A ∪ A = A
• A ∩ A = A
• A ∩ ∅ = ∅
• A ∪ ∅ = A
• A ∩ B ⊆ A
• A ⊆ A ∪ B

Important Notes on Set Operations

• Set operation formula for union of sets is n(A∪B) = n(A) + n(B) − n(A∩B) and set operation formula for intersection of sets is n(A∩B) = n(A)+n(B)−n(A∪B).
• The union of any set with the universal set gives the universal set and the intersection of any set A with the universal set gives the set A.
• Union, intersection, difference, and complement are the various operations on sets.
• The complement of a universal set is an empty set U′ = ϕ. The complement of an empty set is a universal set ϕ′ = U.

Related Topics on Set Operations

• Superset
• Power Set
• Finite and Infinite Sets

What are Set Operations in Set Theory?

Set operations are the operations that are applied on two or more sets to develop a relationship between them. There are four main kinds of set operations.

What are the Different Set Operations?

There are four main kinds of set operations which are:

• Union of sets
• Intersection of sets
• Complement of a set
• Difference between sets/Relative Complement

How Do We Use Set Operations in Real Life?

A set is a collection of elements. Some real-life examples of sets are a list of all the states in a country, a list of all shapes in geometry, list of all whole numbers from 1 to 100. We can determine the common regions using the intersection set operation.

How do you Solve Set Operations Problems?

To solve set operation problems we use a Venn diagram to represent the relationship between the sets and apply the set operations formula for union, intersection, difference, or complement of a set.

Which of the Set Operations are Commutative and not Commutative?

Union and Intersection of sets are set operations that are commutative whereas the set difference is not commutative.

What are the Set Operations Symbols?

There are different symbols used for different set operations are referred as set notations. For the union of sets, we use '∪', for the intersection of sets, we use '∩', for the difference of sets, we use ' - ', and for the complement of a set A, we write it as A' or A^c.

How do you Find the Difference Between the Two Sets?

For any two sets A and B, the difference A - B lists all the elements in set A that are not in set B.

How do you Find the Complement of a Set?

Given the universal set 'U' and set A, the complement of a set A is defined as the set of all elements in the universal set that are not present in set A.

What are the Union and Intersection operations of Sets?

For any two sets A and B, the union is defined as the combination of elements in both set A and B. Intersection of sets gives the common elements in set A and set B.