Types of Sets: Definitions, Symbols, Examples

There are different types of sets in mathematics. Sets are defined as a collection of things whose elements are fixed and cannot be modified. In other words, a set is a collection of data that does not pass from one person to the next, for example, a collection of books, kinder joy toys, a group of stamps and coins, etc.

Types of sets are empty sets, equivalent sets, finite sets, subsets, supersets, universal sets, infinite sets, etc. A set of items can be represented using a variety of notations. A roster form or a set builder form commonly represent sets. Let’s take a closer look at each of these terms.

Set Definition

A set is a collection of well-defined objects. Here, well-defined signifies that it must be clear which object belongs to the particular set and which does not. For example, a collection of young players is not a set as the age range for young players is not given, i.e., it cannot be decided which player is to be considered young, and thus, the objects are not well-defined.

If we take a group of players aged 14 to 18, the age range of the players is given in such a way that it is easy to pick which players should be included and which should be removed. Hence, the objects are well-defined for this set.

The objects or members of each of the following collections form a set.

1. A collection of red colour flowers
2. A collection of whole numbers between \(7\) to \(27\)
3. A collection of all the problems in the mathematics book that are difficult for Preethu to solve
4. A collection of vowels, etc

Types of Sets

As of now, we are well aware of the definition of a set. But, then, we have to find whether a given set has a countable number of elements, whether a given set is empty or not, whether two sets have identical elements, or two given sets have an equal number of elements. For all these queries, we have to know different types of sets. 

The different types of sets are as follows.

1. Finite Set: A set is a finite set if it has a limited number of elements in it, i.e., the number of elements that can be counted. 
For example, \(A{\rm{ = \{ Whole}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{20\} ,}}\) \(B = \left\{ {{\rm{Name}}\,{\rm{of}}\,{\rm{teachers}}\,{\rm{in}}\,{\rm{your}}\,{\rm{school}}} \right\}{\rm{,}}\) and so on.

2. Infinite Set: A set is an infinite set if it has an unlimited number of elements, i.e., the number of elements of such a set can not be counted. If a set is not a finite set, then it will be an infinite set. 
For example, \(Z{\rm{ = \{ }}A\,{\rm{set}}\,{\rm{of}}\,{\rm{all}}\,{\rm{the}}\,{\rm{negative}}\,{\rm{integers\} ,}}\) \(X{\rm{ = \{ }}A\,{\rm{set}}\,{\rm{of}}\,{\rm{whole}}\,{\rm{numbers}}\,{\rm{more}}\,{\rm{than}}\,{\rm{20\} ,}}\) \(H{\rm{ = \{ Stars}}\,{\rm{in}}\,{\rm{the}}\,{\rm{sky\} ,}}\) and so on.

3. The empty Set or the Null Set: The empty set is a set that has no elements. The empty set or null set is expressed by a pair of curly braces with no element written inside them, i.e., \(\left\{ {} \right\}\) represent an empty set.
The empty set or null set can also be represented by the Greek letter \(\left( \phi \right),\) spelt as phi. For example, \(A = \left\{ {{\rm{Whole}}\,{\rm{number}}\,{\rm{less}}\,{\rm{than}}\,{\rm{0}}} \right\}{\rm{,}}\) \(B = \left\{ {{\rm{Natural}}\,{\rm{Number}}\,{\rm{less}}\,{\rm{than}}\,{\rm{1}}} \right\}{\rm{,}}\) \(C = \left\{ {{\rm{Triangles}}\,{\rm{with}}\,{\rm{four}}\,{\rm{sides}}} \right\}{\rm{,}}\) and so on.

4. Singleton Set: If a set contains only one element, then the set is known as a singleton set.
For example, \(P = \left\{ {{\rm{The}}\,{\rm{even}}\,{\rm{prime}}\,{\rm{number}}} \right\};\) Since we know the only even prime number is \(2,\) thus the set \(P\) is a singleton set.

5. Equal Sets: Two sets are equal if the elements of the two sets are the same, i.e. if the elements of both sets are identical. The symbol used for equality of sets is =, i.e., equal to.
For example, Let \(P = \{ 1,2,3,4,5\} ,\) \(Q{\rm{ = \{ Whole}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{6\} }}{\rm{.}}\) Then we can say that set P is equal to set \(Q,\) i.e., Set \(P=\)Set \(Q.\)

6. Equivalent Sets: Two sets are equivalent if the number of elements in both sets is equal. The elements can be the same or different, but each set must contain the same number of elements.
For example, Consider \(P{\rm{ = }}\left\{ {{\rm{Bangalore,}}\,{\rm{Coimbatore,}}\,{\rm{Chennai,}}\,{\rm{Wellington}}} \right\}\) and \(Q = {\rm{\{ Messi,}}\,{\rm{Ronaldo,}}\,{\rm{Ramos,}}\,{\rm{Iniesta\} }}{\rm{.}}\) Here \(P\) and \(Q\) are the equivalent sets because they have an equal number of elements, i.e., in each set \(4\) elements are present.

7. Disjoint Sets: If the two given sets have no common elements, they are called disjoint sets.
For example, \(A = \{ a,b,c,d,e\} \) and \(B = \{ 1,2,3,4,5\} .\) Clearly, sets \(A\) and \(B\) have no element in common, there we can say that set \(A\) and set \(B\) are disjoint sets. Let \(P = {\rm{\{ Set}}\,{\rm{of}}\,{\rm{students}}\,{\rm{of}}\,{\rm{class}}\,{\rm{IV\} }}\) and \(Q = {\rm{\{ Set}}\,{\rm{of}}\,{\rm{students}}\,{\rm{of}}\,{\rm{class}}\,{\rm{VI\} }}{\rm{.}}\) Since no students can be common to the two-class, set \(P\) and set \(Q\) are disjoint sets.

8. Overlapping Sets: If two given sets have at least one element in common, they are overlapping sets. Overlapping sets are also known as joint sets.
If set \(P = \{ 1,2,3,4,5,6,7,8,9\} \) and set \(Q = \{ 7,8,9,10,11,12,13\} ,\) set \(P\) and set \(Q\) are overlapping sets as they have elements \(7, 8\) and \(9\) in common.

9. Power Sets: The set of all the subsets is known as power sets. Thus, the power set of a set \(X\) is the set that contains all the subsets of the set \(X.\) For example, if set \(P = \{ 1,2\} ,\) then the power set of \(X\) will be
\(P(X) = \{ \{ \} ,\{ 1\} ,\{ 2\} ,\{ 1,2\} \} \)

10. Subset: A set \(A\) is said to be the subset of a set \(B,\) if the elements of set \(A\) belong to set \(B.\) The symbol used to represent subset is \( \subset .\) Hence, \(A \subset B.\) For examples, if \(P = {\rm{\{ Delhi,}}\,{\rm{Bangalore\} ,}}\) then the subsets of \(P{\rm{ = \{ \} ,\{ Delhi\} , \{ Bangalore\} ,\{ Delhi,}}\,{\rm{Bangalore\} }}\)

11. Universal Set: A universal set is a set that contains all the elements of the other given sets. For example, \(A = \{ 1,2,3,4\} ,B = \{ 4,5,6,7\} ,C = \{ 6,7,8,9,10\} ,\) then, we can represent universal set as \(U = \{ 1,2,3,4,5,6,7,8,9,10\} .\)
Therefore, \(A \subset U,B \subset U,C \subset U.\)

Solved Examples

Q.1. Which of the following set is an empty set. Explain.
(a) Set of even numbers that are not divisible by \(2\)
(b) Set of counting numbers between \(9\) and \(12\)
Ans: We know that the empty set is a set that has no elements. Now,
(a) Even numbers are those numbers that are divisible by \(2.\) There exist no even number that is not divisible by \(2.\) Thus, this set will be empty.
(b) Let \(A\) be the set with counting numbers between \(9\) and \(12.\) Thus \(A = \{ 10,11\} .\) Hence, this set is not empty as it contains the elements inside the set.

Q.2. State which of the following are finite sets and which are infinite.
(a) A set of integers.
(b) A set of boys in your school
(c) A set of football players in a team
Ans: A set is a finite set if it has a limited number of elements in it. \(A\) set is an infinite set if it has an unlimited number of elements. Now,
(a) A set of integers is an infinite set.
(b) A set of boys in your school is a finite set.
(c) A set of football players in a team is a finite set.

Q.3. State if the pair of sets are equal sets or equivalent sets.
{Letters in the word BOARD} and {Whole numbers less than \(5\)}
{Odd number less than \(10\)} and \(\left\{ {1,3,5,7,9} \right\}\)
Ans: (a) \({\rm{\{ Letters}}\,{\rm{in}}\,{\rm{the}}\,{\rm{word}}\,{\rm{BOARD\} = \{ B, O, A, R, D\} }}\) and \({\rm{\{ Whole}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{5\} }} = \{ 0,1,2,3,4\} \)
Thus, both sets have an equal number of elements inside their set, but the elements are not the same. Hence, they are equivalent sets.
(b) \({\rm{\{ Odd}}\,{\rm{number}}\,{\rm{less}}\,{\rm{than}}\,{\rm{10\} = \{ 1,3,5,7,9\} }}\) and the other given set is \(\{ 1,3,5,7,9\} .\) Thus, we can see that the number of elements and the elements is the same in both sets. Hence, they are equal sets.

Q.4. State which of the following pairs are disjoint sets and which are overlapping sets.
(a) \(A=\){Letters in the word DEHRADUN} and \(B=\){Letters in the word BARCELONA}
(b) \(P=\){Boys with age below \(10\) years} and \(Q=\){Boys with age above \(10\) years}
Ans: (a) \(A = {\rm{\{ Letters}}\,{\rm{in}}\,{\rm{the}}\,{\rm{word}}\,{\rm{DEHRADUN\} }}\) and \(B = {\rm{\{ Letters}}\,{\rm{in}}\,{\rm{the}}\,{\rm{word}}\,{\rm{BARCELONA\} }}{\rm{.}}\) They are overlapping sets as some of the elements are common in both sets.
(b) \(P{\rm{ = \{ Boys}}\,{\rm{with}}\,{\rm{age}}\,{\rm{below}}\,{\rm{10}}\,{\rm{years\} }}\) and \(Q{\rm{ = \{ Boys}}\,{\rm{with}}\,{\rm{age}}\,{\rm{above}}\,{\rm{10}}\,{\rm{years\} }}{\rm{.}}\) They are disjoint sets as none of the elements will be shared in both sets.

Q.5. If a set \(A = \left\{ {1,2,3} \right\},\) find all the subset of \(A.\)
Ans: Set \(A = \left\{ {1,2,3} \right\},\) then the subsets of \(A = \{ \} ,\{ 1\} ,\{ 2\} ,\{ 3\} ,\{ 1,2\} ,\{ 2,3\} ,\{ 1,3\} ,\left\{ {1,2,3} \right\}.\)

FAQs

Q.1. What type of a set is \(\{ \} \)?
Ans: \(\{ \} \) denotes that there are no elements in the given set, i.e., it is an empty set or null set.

Q.2. What is a set? Explain with the help of an example.
Ans: A set is a collection of well-defined objects. Here, well-defined signifies that it must be clear which object belongs to the particular set and which does not. So, for example, a collection of lovely flowers is not a set because the objects (flowers) included in the set are not well-defined, whereas a collection of red flowers is a set.

Q.3. What are the different types of sets?
Ans: 1. Empty set or null set
2. Finite set
3. Infinite set
4. Singleton set
5. Equal sets
6. Equivalent sets
7. Disjoint sets
8. Overlapping sets
9. Power sets
10. Subsets
11. Universal set

Q.4. What is a universal set?
Ans: A universal set is a set that contains all the elements of the other given sets.

Q.5. What are the uses of sets?
Ans: In Mathematics, a set can help us to place well-defined objects under one set.

We hope this detailed article on Types of Set helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.