Set Operations

Set operations are analogous to basic mathematical operations. In mathematics, a set is a finite collection of items, such as numbers, alphabets, or real-world objects. When the need to establish the relationship between two or more sets emerges, we must act quickly. Set operations are introduced at this point.

Set union, set intersection, set complement, and set difference are four of the most common set operations. The numerous set operations, notations for describing sets, how to operate on sets, and their applications in real life will all be covered in this article.

Set Operations

A collection of objects is defined as a set. ‘Elements’ are the objects that make up a set. There are three ways to depict a set. Statement form, roster form, and set builder notation are the three types of notation. Set operations are actions performed on two or more sets in order to establish a relationship between them. Set operations are divided into four categories.

• Union of sets

• Intersection of sets

• Complement of a set

• Difference between sets/Relative Complement

Let’s review the concept of Venn diagrams before moving on to cover the various set operations. A Venn diagram is a logical diagram that depicts the potential relationship between two finite sets.

Basic Set Operations

Let’s go over each set action one by one now that we’ve covered the basics of sets and the Venn diagram. The following are some examples of different set operations:

Union of Sets

A ∪ B (read as A union B) is the set of distinct items that belong to both sets A and B or both sets A and B for two given sets A and B. A ∪ B is given by n(A∪B) = n(A) + n(B) − n(A∩B) Consider the following example to better comprehend the set operation of the union of sets: If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, A ∪ B = {1, 2, 3, 4, 5, 6, 7} is the union of A and B.

Intersection of Sets

The set of common elements that belong to both sets A and B is A∩B (read as A intersection B). n(A∩B) = n(A)+n(B)−n(A∪B),( Consider the following example to better comprehend this set operation of set intersection: A ∩ B = {3, 4} is the intersection of A and B if  A = {1, 2, 3, 4} and B = {3, 4, 5, 7}.

Set Difference

The difference between sets and set action requires subtracting items from a set, which is analogous to the difference between numbers. All the elements that are in set A but not in set B are listed in the difference between sets A and B, marked by A – B. Let’s look at an example to better grasp the set operation of set difference: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, The difference between sets A and B can then be calculated as follows: A – B = {1, 2}.

Complement of Sets

The complement of a set A, indicated as A′ or Ac (read as A complement), is the set of all the elements in the specified universal set(U) that aren’t present in set A. Consider the following example to better understand the set operation of complement of sets: 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}.

Properties of Set Operations

Set operations have qualities that are analogous to basic operations on integers. Set operations have the following important properties:

Commutative Law – The commutative property is defined as follows for any two sets A and B,

A ∪ B = B ∪ A

This indicates that the union of two sets is a commutative set operation

A ∩ B = B ∩ A

This means that intersecting two sets is a commutative set action.

The associative property is defined as,

(A ∪ B) ∪ C = A ∪ (B ∪ C)

As a result, the associative set action of set union is used.

(A ∩ B) ∩ C = A ∩ (B ∩ C)

This indicates that the intersection of sets set operation is associative.

De-Law Morgan’s is a set of rules that governs how things are done. We have (A ∪ B)’ = A’ ∩ B’ and (A ∩ B)’ = A’ ∪ B’ for any two sets A and B, according to De Morgan’s law.

Conclusion

n(A∪B) = n(A) + n(B) − n(A∩B), and n(A∩B) = n(A)+n(B)−n(A∪B).

The universal set is obtained by joining any set to the universal set, and the set A is obtained by intersecting any set A with the universal set.

On sets, there are four operations: union, intersection, difference, and complement.

An empty set, U′ = ϕ, is the complement of a universal set. A universal set, ϕ′ = U, is the complement of an empty set.