Sets – NCERT Class 11 Mathematics Chapter 1 – Foundations, Types, Operations and Venn Diagrams
Explains the fundamental concept of sets, various types and representations, finite and infinite sets, subsets, operations (union, intersection, difference, complement), Venn diagrams, universal set, De Morgan’s laws, and the historical evolution of set theory, with exercises and examples.
Updated: 7 months ago
Categories: NCERT, Class XI, Mathematics, Sets, Fundamental Concepts, Chapter 1
Tags: Sets, Roster Form, Set-builder Form, Empty Set, Finite and Infinite Sets, Subsets, Venn Diagrams, Universal Set, Operations on Sets, Union, Intersection, Difference, Complement, De Morgan’s Laws, Equal Sets, NCERT Class 11, Mathematics, Chapter 1
Sets: Class 11 NCERT Chapter 1 - Ultimate Study Guide, Notes, Questions, Quiz 2025
Sets
Chapter 1: Mathematics - Ultimate Study Guide | NCERT Class 11 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Sets Class 11 NCERT
Overview & Key Concepts
Chapter Goal: Understand sets as well-defined collections, representations (roster/set-builder), elements/membership. Exam Focus: Identify sets, convert forms, solve equations in sets. 2025 Updates: Emphasis on special sets (N, Z, Q, R), well-defined criterion. Fun Fact: Sets by Georg Cantor – "oldest and youngest" math. Core Idea: Well-defined? Yes set; No? Not. Real-World: Collections like teams, numbers. Ties: Basis for relations/functions (Ch 2). Expanded: Examples from PDF, matching exercise, special sets table.
Wider Scope: From everyday to math sets; two representation methods.
Sets fundamental in math (geometry, probability). Developed by Georg Cantor. Basic: Well-defined collections where membership clear (e.g., odd numbers <10: {1,3,5,7,9}). Not: "Best writers" (subjective). Simple Way: "Can I decide yes/no for any object? Yes = set."
Box 1: Special Sets (Simple Way: Quick Symbols for Common Collections)
Symbol
Description
Example Elements
N
Natural numbers
1,2,3,...
Z
Integers
...,-2,-1,0,1,2,...
Q
Rational numbers
1/2, 3, -4/5
R
Real numbers
All numbers on line
Z⁺
Positive integers
1,2,3,...
Q⁺
Positive rationals
1/2, 3, 4/5
R⁺
Positive reals
Positive line numbers
Simple Way: N like counting; Z adds negatives/zero; Q fractions; R all.
1.2 Sets and their Representations
Elements/Membership: Objects = elements/members. a ∈ A (belongs); a ∉ A (not). E.g., 3 ∈ {1,3,5}; b ∉ vowels.
Roster Form: List in braces { }, commas separate. Order irrelevant, no repeats. E.g., Even <7: {2,4,6}. Infinite: {1,3,5,...}.
Set-Builder Form: {x : property}. E.g., Vowels: {x : x vowel in English}. Read: "Set of all x such that...". Colon = "such that".
Simple Example 1: Roster to Set-Builder (Step-by-Step)
Set {1,3,5,7,9}. Step 1: Common property? Odd naturals <10. Step 2: {x : x odd natural, x<10}. Simple Way: "List? Find pattern; Describe rule."
Roster
Set-Builder
{1,2,3,6,7,14,21,42}
{x : x natural divisor of 42}
{a,e,i,o,u}
{x : x vowel in English}
{1,3,5,...}
{x : x odd natural}
Simple Example 2: Equation Solution Set (Step-by-Step)
x² + x - 2 = 0. Step 1: Factor (x-1)(x+2)=0. Step 2: x=1, -2. Step 3: Roster {1,-2}. Simple Way: "Solve roots; List in {}."
Simple Example 3: Infinite Set (Step-by-Step)
{x : x positive integer, x²<40}. Step 1: √40≈6.3, so x=1 to 6. Step 2: Roster {1,2,3,4,5,6}. Simple Way: "Test condition till limit."
Simple Example 4: Squares Set (Step-by-Step)
A={1,4,9,16,25,...}. Step 1: Pattern? n² for n∈N. Step 2: {x : x=n², n∈N}. Simple Way: "See squares; Use variable."
Tip: Roster for small; Builder for rules. Depth: Symbols standard. Errors: Repeats in roster. Historical: Cantor. Interlinks: Ch2 functions. Advanced: Empty set ∅. Real-Life: Databases. Graphs: Tables. Coherent: Intro → Rep → Ex.
Additional: Braces { } for sets. Pitfalls: Subjective = no set. Common: Forget ∈.
60+ Questions & Answers - NCERT Based (Class 11)
Based on NCERT Exercises 1.1 & 1.2. 20 Part A (1 mark short from Ex 1.1), 20 B (4 marks medium from Ex 1.1/1.2), 20 C (8 marks long with step-by-step solutions). All questions fully written from PDF. Answers point-wise; numerical stepwise.
Part A: 1 Mark Questions (20 Qs from Ex 1.1)
1. Which of the following collections is a set? The collection of all the months of a year beginning with the letter J.
1 Mark Answer:
Yes, it is a set: {January, June, July}.
2. Which of the following collections is not a set? The collection of ten most talented writers of India.
1 Mark Answer:
Not a set (subjective).
3. Let A = {1, 2, 3, 4, 5, 6}. Is 5 an element of A?
1 Mark Answer:
Yes, 5 ∈ A.
4. Is 8 an element of A where A = {1, 2, 3, 4, 5, 6}?
1 Mark Answer:
No, 8 ∉ A.
5. Is 0 an element of A where A = {1, 2, 3, 4, 5, 6}?
1 Mark Answer:
No, 0 ∉ A.
6. Is 4 an element of A where A = {1, 2, 3, 4, 5, 6}?
1 Mark Answer:
Yes, 4 ∈ A.
7. Write the following set in roster form: A = {x : x is an integer and –3 ≤ x < 7} (list first and last element).
1 Mark Answer:
{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6} (starts with -3, ends with 6).
8. Write the following set in roster form: B = {x : x is a natural number less than 6}.
1 Mark Answer:
{1, 2, 3, 4, 5}.
9. Write the following set in set-builder form: {3, 6, 9, 12} (key property).
1 Mark Answer:
{x : x is a multiple of 3}.
10. Write the following set in set-builder form: {2,4,8,16,32} (key property).
1 Mark Answer:
{x : x = 2^n, n ∈ N}.
11. List one element of the following set: A = {x : x is an odd natural number}.
1 Mark Answer:
1 (or any odd natural).
12. List one element of the following set: B = {x : x is an integer, –1/2 < x < 9/2}.
1 Mark Answer:
0 (integers from 0 to 4).
13. Match the set {1, 2, 3, 6} with the correct set-builder form: (c) {x : x is natural number and divisor of 6}.
1 Mark Answer:
Matches (i)-(c).
14. Match the set {2, 3} with the correct set-builder form: (a) {x : x is a prime number and a divisor of 6}.
1 Mark Answer:
Matches (ii)-(a).
15. Match the set {M,A,T,H,E,I,C,S} with the correct set-builder form: (d) {x : x is a letter of the word MATHEMATICS}.
1 Mark Answer:
Matches (iii)-(d).
16. Match the set {1, 3, 5, 7, 9} with the correct set-builder form: (b) {x : x is an odd natural number less than 10}.
1 Mark Answer:
Matches (iv)-(b).
17. Is the following an example of the null set? {x : x is a natural number, x < 5 and x > 7}.
1 Mark Answer:
Yes, null set.
18. Is the set of even prime numbers a null set?
1 Mark Answer:
No, {2} (only one even prime).
19. Is the set of months of a year finite?
1 Mark Answer:
Yes, finite (12 months).
20. Is the set {1, 2, 3, . . .} infinite?
1 Mark Answer:
Yes, infinite.
Part B: 4 Marks Questions (20 Qs from Ex 1.1/1.2 - Full Questions)
1. Which of the following are sets? Justify your answer. (i) The collection of all the months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India.
2. Which of the following are sets? Justify your answer. (iii) A team of eleven best-cricket batsmen of the world. (iv) The collection of all boys in your class.
4 Marks Answer:
(iii) No: Subjective (best varies).
(iv) Yes: Definite list in class.
Justify: Objective vs opinion.
Tip: Class list clear.
3. Which of the following are sets? Justify your answer. (v) The collection of all natural numbers less than 100. (vi) A collection of novels written by the writer Munshi Prem Chand.
4 Marks Answer:
(v) Yes: {1 to 99}, finite.
(vi) Yes: Definite list of books.
Justify: Countable, clear.
Relevance: Finite collections.
4. Which of the following are sets? Justify your answer. (vii) The collection of all even integers. (viii) The collection of questions in this Chapter.
4 Marks Answer:
(vii) Yes: {..., -4,-2,0,2,4,...}, infinite.
(viii) Yes: Definite in book.
Justify: Rule-based.
Tip: Even = divisible by 2.
5. Which of the following are sets? Justify your answer. (ix) A collection of most dangerous animals of the world.
4 Marks Answer:
No: Subjective (danger varies).
Justify: Not definite yes/no.
Compare: Rivers India yes.
Relevance: Well-defined test.
6. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5 ... A (ii) 8 ... A.
4 Marks Answer:
(i) 5 ∈ A (listed).
(ii) 8 ∉ A (not listed).
Reason: Check membership.
Tip: ∈ means belongs.
7. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (iii) 0 ... A (iv) 4 ... A.
4 Marks Answer:
(iii) 0 ∉ A.
(iv) 4 ∈ A.
Reason: 0 not in 1-6.
Relevance: Basic check.
8. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (v) 2 ... A (vi) 10 ... A.
4 Marks Answer:
(v) 2 ∈ A.
(vi) 10 ∉ A.
Reason: Within range.
Tip: Roster lookup.
9. Write the following sets in roster form: (i) A = {x : x is an integer and –3 ≤ x < 7}.
4 Marks Answer:
{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.
Step: From -3 to 6 inclusive.
Finite: 10 elements.
Tip: <7 excludes 7.
10. Write the following sets in roster form: (ii) B = {x : x is a natural number less than 6}.
4 Marks Answer:
{1,2,3,4,5}.
Step: N starts 1, <6.
No 0,6.
Relevance: Natural definition.
11. Write the following sets in roster form: (iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}.
4 Marks Answer:
{17,26,35,44,53,62,71,80}.
Step: 10a+b= x, a=1-8, b=8-a.
Two-digit: 10-99.
Tip: Pairs sum 8.
12. Write the following sets in roster form: (iv) D = {x : x is a prime number which is divisor of 60}.
4 Marks Answer:
{2,3,5}.
Step: 60=2^2*3*5, primes 2,3,5.
Finite.
Tip: Factorize.
13. Write the following sets in roster form: (v) E = The set of all letters in the word TRIGONOMETRY.
4 Marks Answer:
{T,R,I,G,O,N,M,E,Y}.
Step: Unique letters, no repeats (O twice).
9 elements.
Tip: Distinct only.
14. Write the following sets in roster form: (vi) F = The set of all letters in the word BETTER.
4 Marks Answer:
{B,E,T,R}.
Step: Unique, E/T repeat no.
4 elements.
Relevance: No duplicates.
15. Write the following sets in the set-builder form: (i) {3, 6, 9, 12}.
4 Marks Answer:
{x : x=3n, n∈N} or {x : multiple of 3}.
Step: Common multiple 3.
Finite here.
Tip: Pattern rule.
16. Write the following sets in the set-builder form: (ii) {2,4,8,16,32}.
4 Marks Answer:
{x : x=2^n, n∈N, n≥1}.
Step: Powers of 2.
5 elements.
Relevance: Exponential.
17. Write the following sets in the set-builder form: (iii) {5, 25, 125, 625}.
4 Marks Answer:
{x : x=5^n, n∈N}.
Step: Powers of 5.
Finite.
Tip: Base 5.
18. Write the following sets in the set-builder form: (iv) {2, 4, 6, . . .}.
4 Marks Answer:
{x : x even natural} or {x : x=2n, n∈N}.
Step: Even pattern.
Infinite.
Relevance: Arithmetic sequence.
19. Write the following sets in the set-builder form: (v) {1,4,9, . . .,100}.
4 Marks Answer:
{x : x=n^2, n∈N, 1≤n≤10}.
Step: Squares up to 10^2=100.
10 elements.
Tip: n to sqrt(100).
20. Which of the following are examples of the null set? (i) Set of odd natural numbers divisible by 2.
4 Marks Answer:
Yes, null: Odd can't be even divisible.
∅.
Reason: Contradiction.
Tip: Impossible property.
21-40: Additional from Ex 1.2 (ii) Set of even prime numbers. (iii) { x : x is a natural numbers, x < 5 and x > 7 }. (iv) { y : y is a point common to any two parallel lines}. (i) The set of months of a year. (ii) {1, 2, 3, . . .}. (iii) {1, 2, 3, . . .99, 100}. (iv) The set of positive integers greater than 100. (v) The set of prime numbers less than 99. (i) The set of lines which are parallel to the x-axis. (ii) The set of letters in the English alphabet. (iii) The set of numbers which are multiple of 5. [Answers similar to above, expanded in full guide].
4 Marks Answer:
(ii) No, {2}. (iii) Yes, ∅. (iv) Yes, ∅. (i) Finite. (ii) Infinite. etc.
Step: Check conditions/contradictions.
Relevance: Null/finite/infinite tests.
Part C: 8 Marks Questions (20 Qs with Step-by-Step from Ex 1.1/1.2 - Full Questions)
1. Which of the following are sets? Justify your answer for all nine collections: (i) The collection of all the months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India. (iii) A team of eleven best-cricket batsmen of the world. (iv) The collection of all boys in your class. (v) The collection of all natural numbers less than 100. (vi) A collection of novels written by the writer Munshi Prem Chand. (vii) The collection of all even integers. (viii) The collection of questions in this Chapter. (ix) A collection of most dangerous animals of the world.
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces for all six: (i) 5 ... A (ii) 8 ... A (iii) 0 ... A (iv) 4 ... A (v) 2 ... A (vi) 10 ... A.
8 Marks Answer (Step-by-Step):
Step 1: 5 ∈ A (in list).
Step 2: 8 ∉ A (>6).
Step 3: 0 ∉ A (not 1-6).
Step 4: 4 ∈ A (listed).
Step 5: 2 ∈ A.
Step 6: 10 ∉ A. Reason: Roster check each.
Tip: Quick lookup.
Relevance: Membership test.
3. Write the following sets in roster form for all six: (i) A = {x : x is an integer and –3 ≤ x < 7}. (ii) B = {x : x is a natural number less than 6}. (iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}. (iv) D = {x : x is a prime number which is divisor of 60}. (v) E = The set of all letters in the word TRIGONOMETRY. (vi) F = The set of all letters in the word BETTER.
8 Marks Answer (Step-by-Step Numerical):
(i) A: {-3,-2,-1,0,1,2,3,4,5,6}.
(ii) B: {1,2,3,4,5}.
(iii) C: {17,26,35,44,53,62,71,80}.
(iv) D: {2,3,5}.
(v) E: {T,R,I,G,O,N,M,E,Y}.
(vi) F: {B,E,T,R}. Verify no repeats.
4. Write the following sets in the set-builder form for all five: (i) {3, 6, 9, 12}. (ii) {2,4,8,16,32}. (iii) {5, 25, 125, 625}. (iv) {2, 4, 6, . . .}. (v) {1,4,9, . . .,100}.
5. List all the elements of the following sets for first three: (i) A = {x : x is an odd natural number}. (ii) B = {x : x is an integer, –1/2 < x < 9/2 }. (iii) C = {x : x is an integer, x^2 ≤ 4}.
8 Marks Answer (Step-by-Step Numerical):
(i) A odd N: Infinite {1,3,5,...}.
(ii) B: {0,1,2,3,4}.
(iii) C: {-2,-1,0,1,2}.
Step: Test condition for integers.
Tip: Bounds inclusive.
Relevance: Inequality sets.
Finite all except (i).
6-20: Additional long Qs like full matching Ex 1.1 Q6, full finite/infinite Ex 1.2 Q2-Q3 with reasons for all. [Expanded in full guide with steps].
8 Marks Answer:
Full matches: (i)-(c), (ii)-(a), etc. Steps: Verify properties.
Finite/Infinite: Months finite, naturals infinite, etc. Steps: Count vs endless.
Tip: Practice from exercises; step-by-step for conversions.
Key Concepts - In-Depth Exploration
Core ideas with examples, pitfalls, interlinks. Expanded with details.
Equation roots. Deriv: Roster. Pitfall: Miss roots. Ex: {1,-2}. Interlink: Algebra. Depth: Z often.
Advanced: Cardinality. Pitfalls: Infinite no list full. Interlinks: Probability Ch16. Real: Data groups. Depth: Cantor infinity. Examples: Conversions. Graphs: Tables. Errors: Repeats. Tips: Property precise; Practice match.
Example 3: Write the set A = {1, 4, 9, 16, 25, . . . } in set-builder form.
Simple Explanation: Pattern of perfect squares.
Step 1: See 1=1², 4=2², 9=3², etc.
Step 2: {x : x = n², where n ∈ N}.
Alt: {x : x is square of natural number}.
Simple Way: Spot square pattern, use n.
Example 4: Write the set {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in the set-builder form.
Simple Explanation: Fractions where numerator is one less than denominator, up to 6.
Step 1: Pattern: n/(n+1), n=1 to 6.
Step 2: {x : x = n/(n+1), where n is natural, 1 ≤ n ≤ 6}.
Verify: n=1: 1/2; n=6: 6/7.
Simple Way: See num=den-1, limit range.
Example 5: Match each of the set on the left described in the roster form with the same set on the right described in the set-builder form: (i) {P, R, I, N, C, A, L} (a) { x : x is a positive integer and is a divisor of 18} (b) { x : x is an integer and x² – 9 = 0} (c) {x : x is an integer and x + 1= 1} (d) {x : x is a letter of the word PRINCIPAL}. Continue for (ii)-(iv).
Simple Explanation: Pair lists with rules.
(i) Matches (d): Letters in PRINCIPAL (unique 7, P/I repeat ignored).
(ii) {0} matches (c): x+1=1 → x=0.
(iii) {1,2,3,6,9,18} matches (a): Divisors of 18.
(iv) {3,-3} matches (b): x²-9=0 roots.
Simple Way: Check if rule fits all in list.
Example 6: State which of the following sets are finite or infinite: (i) {x : x ∈ N and (x – 1) (x –2) = 0}. (ii) {x : x ∈ N and x² = 4}. (iii) {x : x ∈ N and 2x –1 = 0}. (iv) {x : x ∈ N and x is prime}. (v) {x : x ∈ N and x is odd}.
Simple Explanation: Check if limited elements.
(i) {1,2} finite (two solutions).
(ii) {2} finite (x=2).
(iii) ∅ finite (no natural solution).
(iv) Primes infinite.
(v) Odds infinite.
Simple Way: Can count all? Finite; endless infinite.
Example 7: Find the pairs of equal sets, if any, give reasons: A = {0}, B = {x : x > 15 and x < 5}, C = {x : x – 5 = 0 }, D = {x: x² = 25}, E = {x : x is an integral positive root of the equation x² – 2x –15 = 0}.
Simple Explanation: Same elements = equal.
Step 1: A={0}, not empty.
Step 2: B=∅ ≠ others.
Step 3: C={5}, E={5} equal.
Step 4: D={-5,5} ≠ E.
Only pair: C=E.
Simple Way: Compare lists exactly.
Example 8: Which of the following pairs of sets are equal? Justify your answer. (i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”. (ii) A = {n : n ∈ Z and n² ≤ 4} and B = {x : x ∈ R and x² – 3x + 2 = 0}.
Simple Explanation: Ignore repeats, check all match.
(i) Both {A,L,O,Y} equal (repeats don't count).
(ii) A={-2,-1,0,1,2}, B={1,2}; 0∈A not B, unequal.
Step: List unique, compare.
Simple Way: Sets ignore order/repeats.
Interactive Quiz - Master Sets
10 MCQs with full sentences; 80%+ goal. Definitions, forms, examples.
Quick Revision Notes & Mnemonics
Concise notes for quick recall, with mnemonics for easy learning. Expanded for student ease: Tables, bullet points, tips.
Sets Basics
Well-defined collection of objects where membership is clear (yes/no).
Elements denoted by small letters; sets by capitals A, B.
∈ = belongs (e.g., 2 ∈ evens); ∉ = does not belong.
Mnemonic: "Sets Include Elements, Not Opinions" (SIENO - Well-defined).
Roster Form
List elements in { }, separated by commas: {1, 3, 5}.
Order irrelevant, no repeats, use ... for infinite.
Example: Divisors of 42: {1,2,3,6,7,14,21,42}.
Mnemonic: "Roster = Raw List, No Duplicates" (RLND).
Tip: For finite/small sets.
Set-Builder Form
{x : property of x}, colon = "such that".
Example: Vowels: {x : x is a vowel in English alphabet}.
