A set is a collection of objects, termed elements. The objects of the sets can be anything from people, districts, countries, or any other possibilities. Application of sets is used in statistics, Boolean algebra, and probability. This chapter discusses the different types of sets, set operations, and properties of sets. 

Types of Sets

Sets are classified based on the nature of the elements present in them. The various types of sets are as follows: 

• Empty set

A set that contains no elements is called an empty set. It is also termed as the null set or the void set. The empty set can be represented using ‘Phi symbol’ (Φ) or empty brackets {}.

• Universal set

According to set theory, a universal set is a set that contains all the elements of the subsets. The universal set is represented using the symbol ‘U’.

Consider set A = {1,2} and set B = {3,4,5}, the universal set U will have elements from both sets A and B. Hence U = {1,2,3,4,5}.

• Finite set

A set that contains a finite number of elements is called a finite set.

Examples of finite set: 

1. A = {1,2,3}. The number of elements in set A is three [n(A)=3]. Hence it is a finite set. 
2. B = {Number of states in India}. The number of elements in set B will be finite [n(B)=a finite number]. Hence it is a finite set.

• Infinite set

A set that does not contain a finite number of elements or a set in which the number of elements is infinite is called an infinite set.

Examples of infinite set: 

1. A = {All natural numbers}. 

Set A = {1,2,3,4,………….}

The number of elements in A, n(A) is infinite. Hence it is an infinite set. 

1. B = {All integers}. 

The number of elements in set B is infinite. Therefore it is an infinite set.

• Subsets

A set A is said to be a subset of a set B if every element of A is also an element of B. It is denoted with the symbol ‘⊂’. If ‘A’ is a subset of ‘B’, it is denoted as A ⊂B.

Examples of subset:

1. Let’s consider two sets, set A = {1,2} and set B = {1,2,3,4,5}. Every element of set ‘A’ is also an element of set ‘B’. Hence A is the subset of B.
2. Let’s consider two sets, set A = {1,2} and set B = {3,4,5,6,7}. The element of set ‘A’ is not an element of set ‘B’. Hence A is not the subset of  B.

Set Operations

The types of set operation are as follows:

• Union of sets

The union of the sets ‘A’ and ‘B’ is denoted as AB. The union of the sets will have all the elements from both the sets and the common elements should be considered only once. 

Example for union of sets: 

1. Let’s consider two sets, set A = {a.b.c} and set B = (c,d,e}.

AB = {a,b,c,d,e}. The element ‘c’ is present in both the sets and hence it is considered once for the union.

• Intersection of sets

The intersection of the sets ‘A’ and ‘B’ is denoted as AB. The intersection of two sets A and B will have only the common elements of sets, A and B.

Example for intersection of sets: 

1. Let’s consider two sets, set A = {a.b.c} and set B = (c,d,e}.

AB = {c}. The element ‘c’ is present in both the sets and hence it is considered for the intersection.

1. Let’s consider two sets, set A = {1,2,3} and set B = (1,2,4,5,6}.

AB = {1,2}. The elements ‘1 and 2’ are present in both the sets and hence it is considered for the intersection.

• Difference of sets

The difference of the set ‘A’ and ‘B’ is denoted as A-B, and the difference of set ‘B’ from set ‘A’ is denoted as B-A. In set theory, A-B≠B-A. A-B will have the elements that belong to set A but not to set B. Likewise, B-A will have the elements that belong to set B but not to set A. 

Example for intersection of sets: 

1. Let’s consider two sets, set A = {1,2,3,4} and set B = (3,4,5,6}.

A-B = {1,2} since the elements 1 and 2 belong to set A but not to set B

B-A = {5,6} since the elements 5 and 6 belong to set B but not set A.

Properties of set 

A few properties or laws based on the set operations are listed below: 

Conclusion 

Set theory is used through maths, and it is the basis for many subfields of maths such as statistics and probability. The operations and properties of set theory have been used widely for many applications.