What is a Set? Types, Properties & Examples

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The sets are the well-defined collection of objects and elements in NCERT Class 11 Maths. The concept of sets is an integral component in the mathematical classification and organization of data.

Set deals with the properties and operations of the collection of elements for data analysis.
Georg Cantor, a German mathematician, developed the set theory in Maths.
A set is named and represented using a capital letter.
Every item in a set is referred to as an element.
The elements in the finite set are represented using curly brackets {…}.
Elements can not be repeated but arranged in a specific order.
The number of elements in the finite set is called a cardinal number.
For instance, set A is a collection of the first five prime numbers, then A = {2, 3, 5, 7, 11}.
The collection of numbers and the school bag of children are all examples of sets.
Keeping sets of books and notebooks separately in a bag.

Table of Content

What is a Set?
Representation of Sets
Types of Sets
Venn Diagram of Sets
Operations on Sets
Sets Formulas
Properties of Sets
Important Topics for JEE Main
Things to Remember
Sample Questions

Key Terms: What is a Set, Sets, Elements, Cardinal Number, Universal Set, Set Builder Form, Roster Form, Singleton Set, Disjoint Sets, Equal Sets, Union of Sets, Subsets, Power of Set

What is a Set?

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Sets are organized collections of objects or elements. It is composed of elements which could be numbers, letters, shapes, etc. The elements of a set do not change from person to person.

A set is represented by a capital letter, for instance, A = {...}.
Elements are represented using a curly bracket in a set such as A = {1, 2, 3, 4, 5}.
All the elements of a set should be interrelated to each other and should share a common property.
It is used to represent collection of data and bulk of data.
Sets in Maths can be any sort of organized data or objects.

Example of What is a Set?

Example 1: Consider the case of shopping mall where different categories of things are divided in separate of shops. It indicates that shoes is on one floor and makeup is on another floor.

Example 2: Find A U B and A ⋂ B and A – B. If A = {a, b, c} and B = {c, d}.

Ans. A = {a, b, c} and B = {c, d}

A U B = {a, b, c}

A ⋂ B = {c, d} and

A – B = {a, b}

Example 3: There is an organized collection of natural numbers, prime numbers, etc in Mathematics. For instance, consider a set A of the first 10 natural numbers, then, it would be represented as

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The video below explains this:

Set in One Shot Detailed Video Explanation:

What are Elements of a Set?

Elements of a set are the objects or items present in a set. In a given set, the elements are enclosed in curly brackets {...} separated by commas.

Elements in a set are denoted by the symbol '∈'.
For instance, if 2 is an element of Set A, then, 2 ∈ A.
If an element is not a part of the given set, then it is denoted by the symbol '∉'.
For instance, if 3 is not an element of Set A, then, 3 ∉ A.

Example of What are Elements of a Set?

Example: Consider B = {1, 2, 3, 4, 5,6 }. Since a set is usually represented by the capital letter. Thus, B is the set and 1, 2, 3, 4, 5,6 are the elements of the set.

Set in Maths

Set in Maths

Cardinal Number of a Set

Cardinal Numbers of a set refer to the total number of elements in the set. It is also known as order of sets. Order of sets describes the size of a set.

The size of a set can be finite or infinite depending upon the type of sets.

Example of Cardinal Number of a Set

Example: For instance, Set A represents the natural even numbers less than, then the set will be A = {2, 4, 6, 8}.

Thus, the Cardinal Number of the set will be n(A) = 4.

Read More:

Chapter Related Concept
Set Formula	Set Theory Symbols	Data Sets
Union and Intersection of Sets of Cardinal Numbers	SET Theory	Reflexive Relation

Representation of Sets

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The representation of sets differs in the way in which the elements are listed. The three set notations or representations used for representing sets are as follows:

Semantic Form

Semantic form of set notation describes a statement to show what are the elements of a set.

Example of Semantic Form

Example: Set A is the list of the first five even numbers.

Roster Form

Roster Form is the form in which elements are listed in a set. Elements in the set are separated by a comma and enclosed within the curly braces. The elements of a set can be finite as well as infinite, thus, their representation will be as follows:

Finite Roster Form of Sets: Set A = {2, 3, 5, 7, 11} (First Five Prime Numbers)
Infinite Roster Form of Sets: Set A = {2, 4, 6, 8 ....} (Multiples of 2)

Example of Roster Form

Example: The set of vowels in roster form is represented as A = {a, e, i, o, u}.

Set Builder Form

Set Builder Form has a certain rule or a statement that describes the common feature of all the elements of a set. It uses a vertical bar in its representation, and a text describing the character of the elements of the set.

Example of Set Builder Form

Example: For instance, A = { k | k is an odd number, k ≤ 30}.

The statement says, all the elements of set A are odd numbers that are less than or equal to 30.
In a few cases, a ":" is used in place of the "|".

Types of Sets

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Sets in maths can be classified into various categories depending on the elements included. The types of sets are as follows:

Singleton Set

The set in which only one element is present is called a Singleton set or Unit set.

Example of Singleton Set

Example: Set A = { k | k is an integer between 3 and 5}. Thus, the set will be represented as A={2}.

Finite Set

The set in which finite elements or countable elements are present is called the Finite set.

Example of Finite Set

Example: A set of vowels in the alphabet. The set will be expressed as A = {a, e, i, o, u}.

Infinite Set

The set in which infinite elements or uncountable elements are present is called the Infinite set.

Example of Infinite Set

Example: A set of numbers divisible by 1. It will be expressed as A = {1,2,3,4,5,6,7,8,9,……..}.

Empty/Null Set

The set which doesn’t contain any element is called an Empty/Null set. An empty set is denoted by { } or Ø.

Example of Empty/Null Set

Example: A set of apples in the basket of oranges is an example of an empty set as there will be no apples in the grape basket.

Equal Sets

The two different sets which contain the same element are called Equal Sets.

Example of Equal Sets

Example: A = {1,2,3} and B = {1,2,3} Thus, Set A and B are equal sets and are denoted by 'A=B'

Unequal Sets

The two different sets that have at least one different element are called Unequal Sets.

Example of Unequal Sets

Example: A= {1,2,3} and B = {1,2,4}. Thus, Set A and B are unequal sets and are denoted by 'A ≠B'.

Disjoint Sets

The two sets which contain no same element are called the Disjoint Sets.

Example of Disjoint Sets

Example: A = {1,2,3} and B = {4,5,6}. Set A and B are disjoint sets as they have all different elements.

Superset and Subset

If every element of set A is present in set B, then set A is a subset of set B (A ⊆ B) and B is the superset of set A(B ⊇ A).

Example of Supeset and Subset

Example: A = {4,8,6} and B = {0,1,2,3,4,5,6,7,8,9}. A ⊆ B, since all the elements in set A are present in set B.

B ⊇ A denotes that set B is the superset of set A.

Power Set

Power set is a set of all the subsets that a set could contain.

Example of Power Set

Example: Set A = {3,4}. Power set of A is = {{∅}, {3}, {4}, {3,4}}.

Universal Set

A universal set is the collection of all the elements of a particular subject. It is denoted by 'U'.

Example of Universal Set

Example: Let U = {List of Integers}. Here, a set of natural numbers, even numbers, and odd numbers are all subsets of this universal set

Venn Diagram of Sets

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Venn Diagrams refer to the pictorial representation of sets in which each set is represented as a circle. The elements of a set are present within the circle.

The universal sets are represented usually by a rectangle and its subsets are represented by circles.
Most of the relationships between sets can be represented through Venn diagrams.
It uses overlapping of a circle or other shapes to depict relationship between other sets.
It is also known as set diagram or logical diagram.
The common area in the venn diagram is called union of sets.

Example of Venn Diagram of Sets

Example: In the given Venn Diagram,

Set A = {1, 2, 3}
Set B = {3, 5, 7}
Thus, the common element of Set A and Set B is 3.

Venn Diagram of Sets

Operations on Sets

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Operations on sets include the various mathematical operations that are applicable to sets such as union, intersections, difference, and the complement of a set, etc.

The various set operations are as follows:

Union of Sets

Union of sets lists the elements in set A and set B or the elements in both set A and set B.

Example of Union of Sets

Example: {3,4} ∪ {1, 4} = {1, 3, 4}. It is denoted as “A U B

Intersection of Sets

Intersection of sets lists the common elements in set A and set B.

Example of Intersection of Sets

Example: {3,4} ∪ {1, 4} = {4}. It is denoted as “A ∩ B

Set Difference

Set difference is the list of elements in set A that are not present in set B.

Example of Set Difference

Example: {3,4} - {1, 4} = {3}. It is denoted as “A - B.

Set Complement

Set complement is the list of all elements present in the Universal set except the elements present in set A

It is denoted as A’.
A' is denoted as U - A, which is the difference between the universal set elements and set A.

Cartesian Product of Sets

If set A and set B are two sets, then the cartesian product of A and B will be a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. Cartesian Product of Sets is denoted by A × B.

A × B = {(a, b) : a ∈ A and b ∈ B}.

Example of Cartesian Product of Sets

Example: Consider that Set A = {1,2,3} and set B = {Bat, Ball}. Thus, A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}.

Read More: Sets Important Questions

Sets Formulas

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Some important Sets Formulas for any three sets P, Q, and R are listed as follows:

n ( P ∪ Q ) = n(P) + n(Q) – n ( P ∩ Q)
If P ∩ Q = ∅, then n ( P ∪ Q ) = n(P) + n(Q)
n( P – Q) + n( P ∩ Q ) = n(P)
n( Q – P) + n( P ∩ Q ) = n(Q)
n( P – Q) + n ( P ∩ Q) + n( Q – P) = n ( P ∪ Q )
n ( P ∪ Q ∪ R ) = n(P) + n(Q) + n(R) – n ( P ∩ Q) – n ( Q ∩ R) – n ( R ∩ P) + n ( P ∩ Q ∩ R)

Read More:

Class 11 Mathematics Related Concepts
Group Theory	Types of Functions	Value of log Infinity
Relations and Functions	Function Notation Formula	Geometric Mean

Properties of Sets

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Sets also have properties such as the associative property, commutative property, and others just like the numbers. The six important properties of sets for three sets A, B, and C are as follows:

Property	Examples
Commutative Property	A ∪ B = B ∪ A (Union)
Commutative Property	A ∩ B = B ∩ A (Intersection)
Associative Property	( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Union)
Associative Property	( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Intersection)
Distributive Property	A ∪ (B ∩ C) = (A ∪ B) ∩ ( A ∪ C) (Union)
Distributive Property	A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Intersection)
Identity Property	A ∪ φ = A (Union)
Identity Property	A ∩ U = A (Intersection)
Complement Property	A ∪ A’ = U (Union)
Idempotent Property	A ∪ A = A (Union)
Idempotent Property	A ∩ A = A (Intersection)

Important Topics for JEE Main

As per JEE Main 2024 Session 1, important topics included in the chapter set are as follows:

Types of Set
Properties of Set
Representation of Set
Subset
Operations on Sets
Sets Formulas

Some memory based important questions asked in JEE Main 2024 Session 1 include:

A = {1,2,3,..., 10), S be the set of subsets of A and R = {(a,b): a, b ∈ S and a n b ≠ Ø). Then R is
If S = {z: z+iz-iz-1, ze C), then the number of elements in Set S = ?
A set R = (1, 2, 3, 4) is given then find the number of symmetric relation which are not reflexive relation.
A (1.2.3.4), R = (1, 2), (2, 3), (2, 4)), RS and S is an equivalence relation, then the minimum number of elements to be added to R is n. Find the value of n

Things to Remember

Sets are collections of objects whose elements are fixed and cannot be changed.
The elements of a set can be any number, name, or object.
Operations on sets include union, intersection, difference, complement, and cartesian product of a set.
A set is denoted by a capital letter and its elements are added within a curly bracket.
It can be represented in three different forms namely, set-builder form, roster form, and statement form.
Finite Sets, Infinite Sets, Universal Sets, Empty Sets, Singleton Sets, Equal Sets, etc are all types of sets.
Students can prepare the topic from Sets Important Questions: Detailed Solutions & Explanation.
Individuals can study Class 11 Maths NCERT Solutions Chapter 1 from this link.

Previous Years’ Questions

If A and B are non-empty sets such that… (KEAM)
Let A and B be two sets then (A∪B)′… (BITSAT - 1990)
Two finite sets have m and n elements. The total number…
Which of the following sets is a finite set…
For any two sets A and B, A-(A−B) equals… (WBJEE - 2009)
The number of proper subsets of a set having… (COMEDK UGET - 2014)
If A and B are two sets, then A ∩ (A∪B)...
Let S be a set containing n elements and we select two subsets… (BITSAT - 2005)
If A and B are not disjoint sets, then n(A∪B)...
Two sets A and B are as under: A = {(a,b)... (JEE Main - 2018)

Sample Questions

Ques. Write the solution set of the equation x² + x – 2 = 0 in roster form. (3 Marks)

Ans. The given equation is a quadratic equation.

On factorizing the given equation x² + x – 2 = 0, we get

x² + 2x - x - 2 = 0

x (x + 2) -1 (x + 2) = 0

(x + 2) (x - 1) = 0

Thus, x = -2 or x = 1

Therefore, the solution set of the equation x² + x - 2 = 0 in roster form is {1, -2}.

Ques. Write the following set in roster form:
(a) A = {x:x is an integer and -3 x < 7}
(b) B = {x:x is a natural number less than 6} (3 Marks)

Ans. (a) A = {x:x is an integer and -3 x < 7}

Integers are: …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,...

A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

(b) B = {x:x is a natural number which is less than 6}

The natural numbers are 1, 2, 3, 4, 5, 6, 7, .....

B = {1, 2, 3, 4, 5}

Ques. Define Power set. (2 Marks)

Ans. Power set is defined as the set of all subsets of a given set including the Set itself and the null or empty set.

Consider A = {1, 3, 5}

Thus, Power set of A will be P(A) = {{}, {1}, {3}, {5}, {1,3}, {3,5}, {1,5}, {1,3,5}}

Ques. With the use of the properties of sets, prove that for all the sets A and B, A - (AB) = A- B. (3 Marks)

Ans. A - (A \(\bigcap\)B) = A \(\bigcap\)(A\(\bigcap\)B)’ (since A - B = A\(\bigcap\)B’)

= A\(\bigcap\)(A’ U B’) [According to De Morgan’s Law]

= (A \(\bigcap\)A’) U (A\(\bigcap\)B’) [According to the Distributive Law]

= \(\varphi\) U (AB’)

= A \(\bigcap\) B’ = A-B

Hence Proved.

Ques. If A and B are two finite sets such that n(A) = 25, n(B) = 20, and n(A ∪ B) = 30, find n(A ∩ B). (2 Marks)

Ans. Using the formula, n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

= 25 + 20 – 30

= 45 – 30

= 15

Ques. Let X = {Amit, Sima, Amar} be the set of students of Class X, who are in the school cricket team. Let Y = {Sima, Dravid, Aakash} be the set of students from Class X who are on the school basketball team. Find X U Y and X \(\bigcap\) Y and interpret the set. (3 Marks)

Ans. We have, X U Y = { Amit, Sima, Amar, Dravid, Aakash }. This is the set of students from Class X who are on the cricket team or the basketball team or both teams.

X \(\bigcap\) Y = {Sima}.

This is the set of students from class X who will play in both teams.

Ques. Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}, find:
(a) A’ \(\bigcap\) (BC’)
(b) (B-A) U (A-C) (3 Marks)

Ans. We know,

U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}

(a) A’ = {1, 3, 5, 7}

C’ = {3, 5, 6}

B \(\bigcap\) C’ = {3, 5}

A’ U (B \(\bigcap\) C’) = {1, 3, 5, 7}

(b) B-A = {3, 5}

A-C = {6}

(B-A) U (A-C) = {3, 5, 6}

Ques. Given that n(A – B) = 30, n(A ∪ B) = 65 and n(A ∩ B) = 22. What will be the value of n(B)? (3 Marks)

Ans. Using the given formula,

n(A ∪ B) = n(A – B) + n(A ∩ B) + n(B – A)

65 = 30 + 22 + n(B – A)

65 = 52 + n(B – A)

n(B – A) = 65 – 52 = 7

Now, we know that n(B) = n(A ∩ B) + n(B – A)

= 22 + 7

= 15

Ques. What will be the given set in a set-builder form: A = {2, 4, 6, 8, 10, 12, 14}? (2 Marks)

Ans. The set is given as A = {2, 4, 6, 8, 10, 12, 14}

It can be represented in Set-Builder form as follows:

A = {x | x is an even natural number which is less than 15}

Ques. Let U = {x : x ∈ N, x ≤ 9}; A = {x : x is an even number, 0 < x < 10}; B = {2, 3, 5, 7}. Write the set (A U B). (3 Marks)

Ans. Consider U = {x : x ∈ N, x ≤ 9}; A = {x : x is an even number, 0 < x < 10}; B = {2, 3, 5, 7}

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {2, 4, 6, 8}

A U B = {2, 3, 4, 5, 6, 7, 8}

(A U B)’ = {1, 9}

Ques. Determine which of the following sets are equal and equivalent in nature. (3 Marks)

Set A= {2, 4, 16, 8, 12}
Set B= {a, b, c, d, e}
Set C= {c: c ∈ N, c is an even number, c ≤ 12}
Set D = {1, 2, 5, 10}
Set E= {x, y, z}

Ans. Equivalent sets are those sets which have an equal number of elements. On the other hand, equal sets are those which have equal number of elements present in the set.

Equivalent Sets: Set A, Set B, Set C.
Equal Sets: Set A, Set C.

Ques. Determine the types of the sets given below: (3 Marks)

Set A= {a: a is the number divisible by 14}
Set B = {2, 4, 8}
Set C = {k}
Set D= {n, m, o, k}
Set E= ϕ

Ans. The type of sets are as follows:

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

Ques. Find the elements of the sets represented as follows and write the cardinal number of each set. (A) Set A is the first 8 multiples of 10 (B) Set B = {m,n,b,v,c} (C) Set D = {x | x are even numbers between 30 and 50}. (3 Marks)

Ans. The solution are as follows:

(A) Set A = {10,20,30,40,50,60,70,80}. These are multiple of 10.

Since there are 8 elements in the set, cardinal number n (A) = 8

(B) Set B = {m,n,b,v,c}. There are five elements in the set,

Therefore, the cardinal number of set B, n(B) = 5.
(C) Set D = {32,34,36,38,40,42,44,46,48}. These are the even numbers between 20 and 40, which make up the elements of the set C.

Therefore, the cardinal number of set C, n(C) = 9.

Ques. Express the given set in set-builder form: A = {3,6,9,12,15,18}. (3 Marks)

Ans. Given: A = {3,6,9,12,15,18,21}

Using sets notations, we can represent the given set A in set-builder form as,

A = {x | x is an odd natural number less than 15}

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What is a Set? Types, Properties & Examples

What is a Set?

Example of What is a Set?

Set in One Shot Detailed Video Explanation:

What are Elements of a Set?

Example of What are Elements of a Set?

Cardinal Number of a Set

Example of Cardinal Number of a Set

Representation of Sets

Semantic Form

Example of Semantic Form

Roster Form

Example of Roster Form

Set Builder Form

Example of Set Builder Form

Types of Sets

Singleton Set

Example of Singleton Set

Finite Set

Example of Finite Set

Infinite Set

Example of Infinite Set

Empty/Null Set

Example of Empty/Null Set

Equal Sets

Example of Equal Sets

Unequal Sets

Example of Unequal Sets

Disjoint Sets

Example of Disjoint Sets

Superset and Subset

Example of Supeset and Subset

Power Set

Example of Power Set

Universal Set

Example of Universal Set

Venn Diagram of Sets

Example of Venn Diagram of Sets

Operations on Sets

Union of Sets

Example of Union of Sets

Intersection of Sets

Example of Intersection of Sets

Set Difference

Example of Set Difference

Set Complement

Cartesian Product of Sets

Example of Cartesian Product of Sets

Sets Formulas

Properties of Sets

Important Topics for JEE Main

Things to Remember

Previous Years’ Questions

Sample Questions

CBSE CLASS XII Related Questions

1.\[ \int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \, dx \text{ is equal to:} \]

2.Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

3. Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that: \[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.

4.In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.

5.If \(\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0\), then the value of \(x\) is:

6. Find the least value of ‘a’ so that $f(x) = 2x^2 - ax + 3$ is an increasing function on $[2, 4]$.

Similar Mathematics Concepts

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1.
\[ \int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \, dx \text{ is equal to:} \]

2.
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

3.
Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that:
\[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.

4.
In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.

5.
If \(\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0\), then the value of \(x\) is:

6.
Find the least value of ‘a’ so that $f(x) = 2x^2 - ax + 3$ is an increasing function on $[2, 4]$.