What is Sets in Math - Types of Sets, Operations, and Solved Examlpes

Introduction

Mathematicians defined a Set as a collection of distinct objects forming a group. The set can have any group of items, a collection of numbers, a group of different alphabets, days of the week, types of vehicles, and many more. Each item in the set is called an Element of the set. Curly {} brackets are used while representing the set.

A few examples of the set are:

Set A = {1,2,3,4,5}
Set B = {a,e,i,o,u}

Definition of Sets in Mathematics

Mathematicians have defined a Set as a collection of distinct objects. Sets are named using capital letters. Also, in the set theory, a set’s elements can be the people’s names, letters of the alphabet, numbers, shapes, places, etc.

Some standard mathematical examples of Sets

Some standard examples of sets regularly used in mathematics are mentioned below.

Set of natural numbers N= {1,2,3,4,5…}
Set of whole numbers W= {0,1,2,3,4,5…}
Set of integers Z = {…-3,-2,-1,0,1,2,3…}
Set of the rationals Q = {p/q where q is an integer and q is not equal to 0}
Set of the irrationals Q’= {x where x is not a rational number}
Set of real numbers R = QUQ’

The above-mentioned sets are infinite, but there can also be finite sets.

Example of the finite Sets

Set of even natural numbers less than 10: set A = {2,4,6,8}
Set of vowels: set B = {a,e,i,o,u}

Elements of a set

The items present in a set are called the elements of the set. The items in the set are closed in curly brackets. Commas separate the elements. To denote that an element is contained in a set, the symbol ∈ is used; if not, then the ∉ symbol is used.

For example:

Given set: Set A = {2,4,6,8}

Therefore, 2 ∈ Set A

Also, 17 ∉ Set A

Cardinal Number of a Set

A set’s cardinal number or order denotes the total number of elements in the set.

For even natural numbers less than 10, n(A) = 4.

Sets are a collection of unique elements, but each element must be related to another by a unique relation.

For example, if we define a set for “the days of the week,” then each element will represent either of the seven days of the week; it cannot represent any month of the year.

Representation of Sets in Set Theory

Different set notations are used to represent sets in set theory. The major difference between the three set notations is the way in which the elements are listed.

The three different forms of representation of sets in mathematics are listed below:

Semantic form
Roster form
Set builder form

For a better understanding of the sets, refer to the table below.

Representation of set A defined as a set of the first five even natural numbers
Semantic Form	A set of the first five even natural numbers
Roster Form	{2,4,6,8,10}
Set Builder Form	{x∈ N \| x ≤ 10 and x is even}

Semantic Form

Semantic notation describes the set of elements using a statement.
For example, set {1,2,3,4,5} is “the set of first five natural numbers” in the semantic notation.

Roster Form

The roster notation is the most common form to represent sets. Roster form is the form of representation in which the elements of the sets are enclosed in the {} and are separated by commas.
For example, Set A = {1,2,3,4,5},
which is the collection of the first five natural numbers.

Points to note:

In a roster form, the set is independent of the order of the set.
For example, the set of the first five natural numbers can also be defined as {2,5,1,4,3}.
If there is an endless list of elements in a set, then a series of dots is used at the end of the last element to define the rest of the series.
For example, infinite sets are represented as X = {1, 2, 3, 4, 5, 6, 7,…}, where X is the set of natural numbers going to infinity.

Set Builder Form

The set builder notation uses a certain rule or a statement that specifies the common feature of a set’s elements.
For example,
A = { k | k is an even number, k ≤ 10}

The exact definition in roster Form is defined as A= {2,4,6,8,10}
The statement says all the elements of set A are even numbers less than or equal to 20. Sometimes a “:” is used instead of the “|”.

Venn Diagram: Method of Visual Representation of Sets

A pictorial representation of sets is called Venn Diagram. Each set is represented in a pictorial representation, for instance, circles. The elements of a set are represented inside the used representation shape. Sometimes a rectangle which represents the universal set, encloses the circles. The Venn diagram also describes the relation between elements of various sets.

Sets Symbols

Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.

Symbol	Meaning
{}	Symbol of set
U	Universal set
n(X)	Cardinal number of set X
b ∈ A	‘b’ is an element of set A
a ∉ B	‘a’ is not an element of set B
∅	Null or empty set
A U B	Set A union set B
A ∩ B	Set A intersection set B
A ⊆ B	Set A is a subset of Set B
B ⊇ A	Set B is the superset of Set A

Types of Sets

There are different types of sets in mathematical set theory. Each set represents a different type of collection of elements. A few basic types of sets are explained below.

Singleton Sets

A set that consists of only one element is referred to as a singleton or unit set.
For example, Set A = { k | k is an integer between 3 and 5}, which is A = {4}.

Finite Sets

A finite set is a set that has a limited or countable number of elements.
Example, Set B = {k | k is a prime number less than 10}, which is B = {2,3,5,7}

Infinite Sets

An infinite set is a set that contains an unlimited number of elements.
Example: Set A = {Multiples of 5}

Empty or Null Sets

An empty set, also known as a null set, is a set that does not contain any elements. It is denoted by the symbol ‘∅’, read as ‘phi’.
Example: Set X = { } = ∅

Equal Sets

When two sets have exactly the same elements, they are called equal sets.
Example: A = {1,8,4} and B = {1,4,8}. Here, set A and set B are equal sets.
This can be represented as A = B.

Unequal Sets

If two sets have at least one different element, they are considered unequal sets.
Example: A = {1,8,4} and B = {2,5,4}.
Here, set A and set B are unequal sets. This can be represented as A ≠ B.

Equivalent Sets

Equivalent sets refer to sets with the same number of elements, even though the elements may differ.
Example: A = {1,2,3} and B = {a,b,c}. Here, set A and set B are equivalent sets since n(A) = n(B)

Overlapping Sets

Two sets overlap if there is at least one element common to both sets.
Example: A = {2,4,9} B = {4,7,10}.
Here, element 4 is present in set A and set B. Therefore, A and B are overlapping sets.

Disjoint Sets

Disjoint sets are sets that do not share any common elements. Example: A = {1,2,3,14} B = {5,6,7,11}. Here, set A and set B are disjoint sets.

Subset and Superset

When every element of set A is also present in set B, set A is considered a subset of set B (A ⊆ B), and set B is considered the superset of set A (B ⊇ A).
Example: Consider the sets A = {1,2,3} and B = {1,2,3,4,8,9,12}.

Universal Set

A universal set is the collection of all elements related to a particular subject. It is denoted by the letter ‘U’.
Example: Let U = {The list of all road transport vehicles}.
Here, a set of cars, cycles, and trains are all subsets of this universal set.

Power Sets

The power set of a set is the collection of all possible subsets that the set can contain.
Example: Set A = {1,2,8}. Power set of A is = {∅, {1}, {2}, {8}, {1,2}, {2,8}, {1,8}, {1,2,8}}.

Operations on Sets

Some important operations on sets in set theory include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of set operations is as follows.

OPERATIONS ON SETS
Union of Sets	The union of sets is denoted as A U B. It lists the elements in both sets A and B.	{1, 3} ∪ {1, 4} = {1, 3, 4}
Intersection of Sets	The intersection of sets, denoted by A ∩ B, lists the elements common to both set A and set B.	{1, 2} ∩ {2, 4} = {2}
Set Difference	The set difference, denoted by A – B, lists the elements in set A that are absent in set B.	A = {2, 3, 4} and B = {4, 5, 6}. A – B = {2, 3}
Set Complement	Set complement, denoted by A’, is the set of all elements in the universal set that are not present in set A. In other words, A’ is denoted as U – A, which is the difference in the elements of the universal set and set A	U= {1,2,3,4,5,6,7,8,9} A= {1,3,5,7,9} A’= {2,4,6,8} = U – A
Cartesian Product of Sets	The cartesian product of two sets, denoted by A × B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained.	{1, 3} × {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}

Sets Formulas in Set Theory
Sets find their application in algebra, statistics, and probability. There are some important set theory formulas in set theory. Refer to the table below for the Set formulas.

For any two overlapping sets, A and B
n(A U B) = n(A) + n(B) – n(A ∩ B)
n (A ∩ B) = n(A) + n(B) – n(A U B)
n(A) = n(A U B) + n(A ∩ B) – n(B)
n(B) = n(A U B) + n(A ∩ B) – n(A)
n(A – B) = n(A U B) – n(B)
n(A – B) = n(A) – n(A ∩ B)
For any two sets A and B that are disjoint
n(A U B) = n(A) + n(B)
A ∩ B = ∅
n(A – B) = n(A)

Properties of Sets

Similar to numbers, sets also have associative property, commutative property, etc. There are six important properties of sets.
Given three sets, A, B, and C, the properties for these sets are as follows.

Property of Set	Example
Commutative Property	A U B = B U A A ∩ B = B ∩ A
Associative Property	(A ∩ B) ∩ C = A ∩ (B ∩ C) (A U B) U C = A U (B U C)
Distributive Property	A U (B ∩ C) = (A U B) ∩ (A U C) A ∩ (B U C) = (A ∩ B) U (A ∩ C)
Identity Property	A U ∅ = A A ∩ U = A
Complement Property	A U A’ = U
Idempotent Property	A ∩ A = A A U A = A

Sets: Solved Examples

Example 1. Let A = {1, 2, 7} and B = {3, 4, 5, 6}. Find the union of sets A and B.
Solution: The union of sets A and B, denoted by A ∪ B, is the set that contains all the elements that are in A or B (or both).
In this case, A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

Example 2. Let A = {1, 2, 4} and B = {3, 4, 5, 6}. Find the intersection of sets A and B.
Solution: The intersection of sets A and B, denoted by A ∩ B, is the set that contains all the elements that are common to both A and B.
In this case, A ∩ B = {4}.

Example 3. Let A = {1, 2, 4} and B = {3, 4, 5, 6}. Find the relative complement of set B with respect to set A.
Solution: The relative complement of set B with respect to set A, denoted by A – B, is the set containing all the elements in A but not in B.
In this case, A – B = {1, 2}.

Example 4. Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Determine whether A is a subset of B.
Solution: A is a subset of B if all the elements of A are also elements of B.
In this case, A is not a subset of B since A contains elements (1, 2) absent in B.

Frequently Asked Question on Sets

How are sets defined in mathematics?

Sets are defined as a well-defined collection of objects. The objects can be people's names, letters of the alphabet, numbers, shapes, places, etc.

What are some examples of sets?

Examples of sets include the set of natural numbers (N), the set of whole numbers (W), the set of integers (Z), the set of rational numbers (Q), and the set of real numbers (R).

What are the elements of a set?

The elements of a set are the individual objects or items that belong to the set.

What is the roster form of representing sets?

Roster form represents sets by listing the elements of the set inside curly brackets, separated by commas.

What is the set builder form of representing sets?

Set builder form uses a rule or statement to specify the common feature of the set's elements. It uses a vertical bar and a condition to describe the elements.