Types of Set Operations

Set operations are related to basic mathematical operations. In mathematics, a set is a finite collection of items, such as numbers, alphabets, or any other real-world objects. We occasionally have to establish the relationship between two or more sets due to a necessity. Set operations are introduced at this point.

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

Formulas of set operations

sets allows us to apply the set formulae, in the domains connected to statistics, probability, geometry, and sequencing.

Set formulae have been constructed from set theory and can be used as a quick reference. Let us review the set notation, symbols, definitions, and properties of sets before proceeding with the formula.

• If n(A) and n(B) denote the number of elements in two finite sets A and B respectively, then for any two overlapping sets A and B, n(A∪B) = n(A) + n(B) – n(A⋂B)

• If A and B are disjoint sets, n(A∪B) = n(A) + n(B)

• If A, B and C are 3 finite sets in U then, n(A∪B∪C)= n(A) +n(B) + n(C) – n(B⋂C) – n (A⋂ B)- n (A⋂C) + n(A⋂B⋂C).

Set operations

A set is defined as a group of items. Each item in a set is referred to as an ‘Element.’ There are three ways to represent a set. There are three of them: statement form, roster form, and set builder notation. Set operations are operations that are performed on two or more sets in order to establish a relationship between them. Set operations are classified into four types, which are as follows.

• Union of sets

• Intersection of sets

• Complement of a set

• Difference between sets/Relative Complement

Before we go into the various set operations, let’s review the notion of Venn diagrams, which is essential for understanding set operations. A Venn diagram is a logical diagram that depicts the potential relationship between several finite sets.

Set operation properties

Set operations have features that are comparable to those of fundamental operations on integers. The following are the key properties of set operations:

• 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 the 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

Conclusion

The idea of set serves as a vital aspect of the general mathematics of day-to-day demands. This notion is now applied in many areas of mathematics. In mathematics, additional sets are utilised to define the notions of relations and functions. Also, understanding of sets is required for the study of geometry, sequencing, probability, and so on.