Representation of a Set | Class 11 Math Notes Study Material Download Free PDF

What are the Sets in Mathematics?

Sets are defined as the collection of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category, even though the elements used in sets are all different from each other, they all are similar as they belong to one group. For instance, a set of different outdoor games, say set A= {Football, basketball, volleyball, cricket, badminton} all the games mentioned are different, but they all are similar in one way as they belong to the same group (outdoor games).

The set is denoted as a capital letter, for example, set A, set B, etc., and the elements belonging to the set are denoted as a small letter, and they are kept in curly brackets {}, for example, set A= {a, b, c, d}, as it is clear that a, b, c, d belong to set A, it can be written a ∈ A, do p belong to set A? No. Therefore, it will be written as, p∉ A.

Representation of Sets

Sets can be represented in two ways, one is known as the Roster form and the other is famous as the Set-Builder form, these two forms can be used to represent the same data, but the style varies in both cases.

Roster Form

In Roster Form, the elements are inside {}⇢ Curly brackets. All the elements are mentioned inside and are separated by commas. Roster form is the easiest way to represent the data in groups. For example, the set for the table of 5 will be, A= {5, 10, 15, 20, 25, 30, 35…..}.

Properties of Roster Formrelations of Sets:

  • The arrangement in the Roster form does not necessarily to be in the same order every time. For example, A= {a, b, c, d, e} is equal to A= {e, d, a, c, b}.
  • The elements are not repeated in the set in Roster form, for example, the word “apple” will be written as, A= {a, p, l, e}
  • The Finite sets are represented either with all the elements or if the elements are too much, they are represented as dots in the middle. The infinite sets are represented with dots in the end.

Set-Builder Form

In Set-builder form, elements are shown or represented in statements expressing relations among elements. The standard form for Set-builder, A= {a: statement}. For example, A = {x: x = a3, a ∈ N, a < 9}

Properties of Set-builder form:

  • In order to write the set in Set- builder form, the data should follow a certain pattern.
  • Colons (:) are necessary in Set-builder form.
  • After colon, the statement is to be written.

Order of the Set

The order of the Set is determined by the number of elements present in the Set. For example, if there are 10 elements in the set, the order of the set becomes 10. For finite sets, the order of the set is finite, and for infinite sets, the order of the set is infinite.

Sample Problems

Question 1: Determine which of the following are considered assetsin and which are not.

  1. All even numbers on the number line.
  2. All the good basketball players from class 9th.
  3. The bad performers from the batch of dancers.
  4. All prime numbers from 1 to 100.
  5. Numbers that are greater than 5 and less than 15.

Answer: 

Sets are not those bunches or groups where some quality or characteristic comes in the picture. Therefore,

  1. “All even numbers on the number line” is a set.
  2. “All the good basketball players from class 9th” is not a Set as “good” is a quality which is involved.
  3. “The bad performers from the batch of dancers” cannot be a Set since “bad” is a characteristic.
  4. “All prime numbers from 1 to 100” is a Set.
  5. “Numbers that are greater than 5 and less than 15” is a Set.

Question 2: Represent the following information inSet-Builder the Roster form.

  1. All Natural numbers.
  2. Numbers greater than 6 and less than 3.
  3. All even numbers from 10 to 25.

Answer:

The Roster form for the above information,

  1. Set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11……}
  2. Set B = {} ⇢ Null set, since there are no numbers greater than 6 and less than 3.
  3. Set C = {10, 12, 14, 16, 18, 20, 22, 24}

Question 3: Express the given information in the Set-Builder form.

  1. Numbers that are greater than 10 and less than 20.
  2. All Natural numbers greater than 25.
  3. Vowels in English Alphabet.

Answer: 

The Set-Builder form for the above information,

  1. A = {a: a∈ N and 10 < a < 20}
  2. B = {b: b∈ N and b > 25}
  3. C = {c: c is the vowel of English Alphabet}

Question 4: Convert the following Sets given in Roster form into Set-Builder form.

  1. A = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
  2. B = {2, 4, 6, 8, 10}
  3. C = {5, 7, 9, 11,13, 15, 17, 19}

Answer: 

The Set- builder form for the above Sets,

  1. A = {a: a is a consonant of the English Alphabet}
  2. B = {b: b is an Even number and 2 ≤ b ≤10}
  3. C = {c: c is an odd number and 5 ≤ c ≤ 19}

Question 5: Give an example of the following types of Sets in both Roster form and Set-builder form.

  1. Singular Set.
  2. Finite Set.
  3. Infinite Set.

Solution:

The Examples can be taken as per choice since there can be a infinite number of examples for any of the above Sets,

  • Singular Set

Roster Form: A = {2}

Set- builder form: A= {a: a∈N and 1<a<3}

  • Finite Set

Roster Form: B = {0,1, 2, 3, 4, 5}

Set-builder form: B = {b: b is a whole number and b<6}

  • Infinite Set

Roster Form: C = {2, 4, 6, 8, 10, 12, 14, 16…..}

Set- builder form: C= {c: c is a Natural and Even number}

Question 6: What is the order of the given sets,

  1. A = {7, 14, 21, 28, 35}
  2. B = {a, b, c, d, e, f, g….x, y, z}
  3. C = {2, 4, 6, 8, 10, 12, 14……}

Answer:

The order of the set tells the number of element present in the Set.

  1. The order of Set A is 5 as it has 5 elements.
  2. The order of set B is 26 as the English Alphabet have 26 letters.
  3. The order of set C is infinite as the set has the infinite number of elements.

Question 7: Express the given Sets in Roster form,

  1. A = {a: a = n/2, n ∈ N, n < 10}
  2. B = {b: b = n2, n ∈ N, n ≤ 5}

Answer:

Representing the above Set-builder sets in Roster form,

  1. A = {1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2}
  2. B = {1, 4, 9, 16, 25}

Conclusion

The representation of a set is a fundamental concept in mathematics that allows us to the describe and manipulate collections of the distinct objects or elements. The Sets can be represented in the various forms including the roster form, set-builder form and Venn diagrams. Understanding these different representations helps in the visualizing and solving problems related to the unions, intersections, subsets and other set operations. Mastery of set representation is essential for the students and professionals working in the fields like mathematics, computer science and logic where sets form the basis for the more complex concepts.

FAQs on Representation of a Set

What is the difference between roster form and set-builder form?

The Roster form lists all the elements of the set explicitly while set-builder form describes the properties that characterize the elements of the set.

Can a set have duplicate elements?

No, a set cannot have duplicate elements. By definition all elements in the set are unique.

What does it mean for a set to be finite or infinite?

A finite set has a specific number of the elements whereas an infinite set has an unbounded number of the elements.

How is an empty set represented?

An empty set, which contains no elements is represented by the symbol ∅ or by a pair of braces with the nothing inside {}.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

CBSE Class 11 Maths Syllabus for 2023-24 with Marking Scheme

CBSE syllabus for class 11 Maths is divided into 5 units. The table below shows the units, number of periods and marks allocated for maths subject. The maths theory paper is of 80 marks and the internal assessment is of 20 marks.

No.UnitsMarks
I.Sets and Functions23
II.Algebra25
III.Coordinate Geometry12
IV.Calculus08
V.Statistics and Probability12
Total Theory80
Internal Assessment20
Grand Total100

2025-26 CBSE Class 11 Maths Syllabus

Below you will find the CBSE Class Maths Syllabus for students.

Unit-I: Sets and Functions

1. Sets

Sets and their representations, empty sets, finite and infinite sets, equal sets, subsets, and subsets of a set of real numbers, especially intervals (with notations), universal set, Venn diagrams, union and intersection of sets, difference of sets, complement of a set and properties of complement.

2. Relations & Functions

Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, Cartesian product of the set of reals with itself (upto R x R x R), definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.

3. Trigonometric Functions

Positive and negative angles, measuring angles in radians and in degrees and conversion from one measure to another, definition of trigonometric functions with the help of unit circle, truth of the identity, signs of trigonometric functions, domain and range of trigonometric functions and their graphs, expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications.

Unit-II: Algebra

1. Complex Numbers and Quadratic Equations

Need for complex numbers, especially√−1, to be motivated by the inability to solve some of the quadratic equations. Algebraic properties of complex numbers, Argand plane.

2. Linear Inequalities

Linear inequalities, algebraic solutions of linear inequalities in one variable and their representation on the number line.

3. Permutations and Combinations

The fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.

4. Binomial Theorem

Historical perspective, statement and proof of the binomial theorem for positive integral indices, Pascal’s triangle, simple applications.

5. Sequence and Series

Sequence and series, arithmetic progression (A. P.), arithmetic mean (A.M.),  geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.

Unit-III: Coordinate Geometry

1. Straight Lines

Brief recall of two-dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.

2. Conic Sections

Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three-Dimensional Geometry

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.

Unit-IV: Calculus

1. Limits and Derivatives

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit, limits of polynomials and rational functions trigonometric, exponential and logarithmic functions, definition of derivative relate it to the slope of the tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

Unit-V: Statistics and Probability

1. Statistics

Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.

2. Probability

Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.

Students can also get the syllabus of all the subjects by visiting CBSE Class 11 Syllabus page. Learn Maths & Science in an interactive & fun-loving way with Anand Classes App/Tablet.

Frequently Asked Questions on CBSE Class 11 Maths Syllabus 2025-26

Q1

What is the marks distribution for internals and theory exams according to the CBSE Maths Syllabus for Class 11?

The marks distribution for internals is 20 marks and the theory exam is 80 marks based on the CBSE Class 11 Maths Syllabus.

Q2

Which is the most important chapter in the CBSE Class 11 Maths Syllabus?

The important chapter in the CBSE Class 11 Maths Syllabus is Algebra which is for 25 marks in the overall weightage.

Q3

What are the chapters covered in Unit III of the CBSE Class 11 Maths Syllabus?

The chapters covered in Unit III of the CBSE Class 11 Maths Syllabus are straight lines, conic sections and an introduction to three-dimensional geometry.