Math

Sets are a well-defined collection of objects. Objects that a set contains are called the elements of the set. We can also consider sets as collections of elements that have a common feature. For example, the collection of even numbers is called the set of even numbers.

What is Set?

A well-defined collection of Objects or items or data is known as a set. The objects or data are known as the element. For Example, the boys in a classroom can be put in one set, all integers from 1 to 100 can become one set, and all prime numbers can be called an Infinite set. The symbol used for sets is {…..}. Only the collection of data with specific characteristics is called a set.

Example: Separate out the collections that can be placed in a set.

• Beautiful Girls in a class
• All even numbers
• Good basketball players
• Natural numbers divisible by 3
• Number from 1 to 10

Answer:

Anything that tries to define a certain quality or characteristics can not be put in a set. Hence, from the above given Collection of data. 

The ones that can be a set,

• All even numbers
• Natural numbers divisible by 3.
• Number from 1 to 10

The ones that cannot be a set,

• Beautiful girls in the park
• Good basketball players

Types of Sets in Mathematics

Sets are the collection of different elements belonging to the same category and there can be different types of sets seen. A set may have an infinite number of elements, may have no elements at all, may have some elements, may have just one element, and so on. Based on all these different ways, sets are classified into different types.

The different types of sets are:

Singleton Set	Empty Set
Finite Set	Infinite Set
Equal Set	Equivalent Set
Subset	Power Set
Universal Set	Disjoint Sets

Let’s discuss these various types of sets in detail.

Singleton Set

Singleton Sets are those sets that have only 1 element present in them.

Example: 

• Set A= {1} is a singleton set as it has only one element, that is, 1.
• Set P = {a : a is an even prime number} is a singleton set as it has only one element 2.

Similarly, all the sets that contain only one element are known as Singleton sets.

Empty Set

Empty sets are also known as Null sets or Void sets. They are the sets with no element/elements in them. They are denoted as ϕ.

Example:

• Set A= {a: a is a number greater than 5 and less than 3}
• Set B= {p: p are the students studying in class 7 and class 8}

Finite Set

Finite Sets are those which have a finite number of elements present, no matter how much they’re increasing number, as long as they are finite in nature, They will be called a Finite set.

Example: 

• Set A= {a: a is the whole number less than 20}
• Set B = {a, b, c, d, e}

Infinite Set

Infinite Sets are those that have an infinite number of elements present, cases in which the number of elements is hard to determine are known as infinite sets. 

Example: 

• Set A= {a: a is an odd number}
• Set B = {2,4,6,8,10,12,14,…..}

Equal Set

Two sets having the same elements and an equal number of elements are called equal sets. The elements in the set may be rearranged, or they may be repeated, but they will still be equal sets.

Example:

• Set A = {1, 2, 6, 5}
• Set B = {2, 1, 5, 6}

In the above example, the elements are 1, 2, 5, 6. Therefore, A= B.

Equivalent Set

Equivalent Sets are those which have the same number of elements present in them. It is important to note that the elements may be different in both sets but the number of elements present is equal. For Instance, if a set has 6 elements in it, and the other set also has 6 elements present, they are equivalent sets.

Example:

Set A= {2, 3, 5, 7, 11}

Set B = {p, q, r, s, t}

Set A and Set B both have 5 elements hence, both are equivalent sets.

Subset

Set A will be called the Subset of Set B if all the elements present in Set A already belong to Set B. The symbol used for the subset is ⊆

If A is a Subset of B, It will be written as A ⊆ B

Example:

Set A= {33, 66, 99}

Set B = {22, 11, 33, 99, 66}

Then, Set A ⊆ Set B 

Power Set

Power set of any set A is defined as the set containing all the subsets of set A. It is denoted by the symbol P(A) and read as Power set of A.

For any set A containing n elements, the total number of subsets formed is 2ⁿ. Thus, the power set of A, P(A) has 2ⁿ elements.

Example: For any set A = {a,b,c}, the power set of A is?

Solution:

Power Set P(A) is,

P(A) = {ϕ, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Universal Set 

A universal set is a set that contains all the elements of the rest of the sets. It can be said that all the sets are the subsets of Universal sets. The universal set is denoted as U.

Example: For Set A = {a, b, c, d} and Set B = {1,2} find the universal set containing both sets.

Solution:

Universal Set U is,

U = {a, b, c, d, e, 1, 2}

Disjoint Sets

For any two sets A and B which do have no common elements are called Disjoint Sets. The intersection of the Disjoint set is ϕ, now for set A and set B A∩B =  ϕ. 

Example: Check whether Set A ={a, b, c, d} and Set B= {1,2} are disjoint or not.

Solution:

Set A ={a, b, c, d}
Set B= {1,2}

Here, A∩B =  ϕ

Thus, Set A and Set B are disjoint sets.

Also, Check

• Set Theory
• Set Theory Symbols
• Relations and Functions
• Representation of a Set
• Operations on Sets

Summarizing Types of Set

There are different types of sets categorized on various parameters. Some types of sets are mentioned below:

Set Name	Description	Example
Empty Set	A set containing no elements whatsoever.	{}
Singleton Set	A set containing exactly one element.	{1}
Finite Set	A set with a limited, countable number of elements.	{apple, banana, orange}
Infinite Set	A set with an uncountable number of elements.	{natural numbers (1, 2, 3, …)}
Equivalent Sets	Sets that have the same number of elements and their elements can be paired one-to-one.	Set A = {1, 2, 3} and Set B = {a, b, c} (assuming a corresponds to 1, b to 2, and c to 3)
Equal Sets	Sets that contain exactly the same elements.	Set A = {1, 2} and Set B = {1, 2}
Universal Set	A set containing all elements relevant to a specific discussion.	The set of all students in a school (when discussing student grades)
Unequal Sets	Sets that do not have all the same elements.	Set A = {1, 2, 3} and Set B = {a, b}
Power Set	The set contains all possible subsets of a given set.	Power Set of {a, b} = { {}, {a}, {b}, {a, b} }
Overlapping Sets	Sets that share at least one common element.	Set A = {1, 2, 3} and Set B = {2, 4, 5}
Disjoint Sets	Sets that have no elements in common.	Set A = {1, 2, 3} and Set B = {a, b, c}
Subset	A set where all elements are also members of another set.	{1, 2} is a subset of {1, 2, 3}

Solved Examples on Types of Sets

Example 1: Represent a universal set on a Venn Diagram.

Solution:

Universal Sets are those that contain all the sets in it. In the below given Venn diagram, Set A and B are given as examples for better understanding of Venn Diagram.

Example:

Set A= {1,2,3,4,5}, Set B = {1,2, 5, 0}

U= {0, 1, 2, 3, 4, 5, 6, 7}

Example 2: Which of the given below sets are equal and which are equivalent in nature?

• Set A= {2, 4, 6, 8, 10}
• Set B= {a, b, c, d, e}
• Set C= {c: c ∈ N, c is an even number, c ≤ 10}
• Set D = {1, 2, 5, 10}
• Set E= {x, y, z}

Solution:

Equivalent sets are those which have the equal number of elements, whereas, Equal sets are those which have the equal number of elements present as well as the elements are same in the set.

Equivalent Sets = Set A, Set B, Set C.

Equal Sets = Set A, Set C.

Example 3: Determine the types of the below-given sets,

•  Set A= {a: a is the number divisible by 10}
• Set B = {2, 4, 6}
• Set C = {p}
• Set D= {n, m, o, p}
• Set E= ϕ

Solution:

From the knowledge gained above in the article, the above-mentioned sets can easily be identified.

• Set A is an Infinite set.
• Set B is a Finite set
• Set C is a singleton set
• Set D is a Finite set
• Set E is a Null set

Example 4: Explain which of the following sets are subsets of Set P,

Set P = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

• Set A = {a, 1, 0, 2}
• Set B ={0, 2, 4}
• Set C = {1, 4, 6, 10}
• Set D = {2, 20}
• Set E ={18, 16, 2, 10}

Solution:

• Set A has elements a, 1, which are not present in the Set P. Therefore, set A is not a Subset.
• Set B has elements which are present in set P, Therefore, Set B ⊆ Set P
• Set C has 1 as an extra element. Hence, not a subset of P
• Set D has 2, 20 as element. Therefore, Set D ⊆ Set P
• Set E has all its elements matching the elements of set P. Hence, Set E ⊆ Set P.

FAQs on Types of Sets

What are sets?

Sets are well-defined collections of objects. 

Example: The collection of Tata cars in the parking lot is a set.

What are Sub Sets?

Subsets of any set are defined as sets that contain some elements of the given set. For example, If set A contains some elements of set B set A is called the subset of set B.

How many types of sets are present?

Different types of sets used in mathematics are 

• Empty Set
• Non-Empty Set
• Finite Set
• Infinite Set
• Singleton Set
• Equivalent Set
• Subset
• Superset
• Power Set
• Universal Set

What is the difference between, ϕ and {ϕ}?

The difference between ϕ and {ϕ} is

• ϕ = this symbol is used to represent the null set, therefore, when only this symbol is given, the set is a Null set or empty set.
• {ϕ}= In this case, the symbol is present inside the brackets used to denote a set, and therefore, now the symbol is acting like an element. Hence, this is a Singleton set.

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