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: 4 days ago
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.
- Expanded Content: Well-defined examples, roster notes (order immaterial, no repeats), set-builder property.
1.1 Introduction
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."
Simple Example 5: Fraction Set (Step-by-Step)
{1/2,2/3,3/4,4/5,5/6,6/7}. Step 1: Pattern? n/(n+1), n=1 to 6. Step 2: {x : x=n/(n+1), n∈N, 1≤n≤6}. Simple Way: "Numerator = denominator -1; Bound range."
Matching Example (Simple Way: Match Description to Form)
| Roster Left | Set-Builder Right | Match |
|---|---|---|
| {P,R,I,N,C,A,L} | {x : letter in PRINCIPAL} | (i)-(d) |
| {0} | {x : x+1=1} | (ii)-(c) |
| {1,2,3,6,9,18} | {x : positive divisor of 18} | (iii)-(a) |
| {3,-3} | {x : x²-9=0} | (iv)-(b) |
Simple Way: "Roster lists; Builder rules. Match property."
Summary
- Sets: Well-defined collections. Roster: List { }; Builder: {x : property}. Use ∈/∉ for membership.
- Special: N,Z,Q,R. Examples: Convert forms, solve sets.
Why This Guide Stands Out
Math-focused: Forms conversion, equation sets, matching. Free 2025 with steps.
Key Themes & Tips
- Aspects: Definition, representation, examples.
- Tip: "Well-defined? Rule or list." Practice conversions.
Exam Case Studies
Identify sets, convert {1,3,5} to builder, match roster-builder.
Project & Group Ideas
- Classify class objects as sets.
- Create infinite sets examples.
Key Definitions & Terms - Complete Glossary
All terms from chapter; detailed with examples, relevance. Expanded: 15+ terms with depth.
Set
Well-defined collection. Relevance: Basis math. Ex: {1,3,5}. Depth: Membership clear.
Well-Defined
Definite belonging. Relevance: Set criterion. Ex: Rivers India yes; Best writers no. Depth: Objective.
Element/Member
Object in set. Relevance: ∈/∉. Ex: 3 ∈ primes. Depth: Synonymous.
Belongs To (∈)
a ∈ A. Relevance: Membership. Ex: a ∈ vowels. Depth: Greek epsilon.
Does Not Belong (∉)
b ∉ A. Relevance: Exclusion. Ex: b ∉ vowels. Depth: Not in set.
Roster Form
List in { }. Relevance: Finite easy. Ex: {2,4,6}. Depth: Order immaterial, no repeats.
Set-Builder Form
{x : property}. Relevance: Infinite/concept. Ex: {x : x even}. Depth: Colon "such that".
Natural Numbers (N)
1,2,3,... Relevance: Counting. Ex: Divisors. Depth: Positive integers start 1.
Integers (Z)
...,-1,0,1,... Relevance: Whole. Ex: Solutions. Depth: From zero.
Rational (Q)
p/q, q≠0. Relevance: Fractions. Ex: 1/2. Depth: Terminable/repeating.
Real (R)
All line numbers. Relevance: Complete. Ex: √2. Depth: Irrational too.
Positive Integers (Z⁺)
1,2,3,... Relevance: Subset. Ex: Divisors. Depth: No zero/neg.
Solution Set
Roots in { }. Relevance: Equations. Ex: {1,-2}. Depth: Roster usually.
Infinite Set
Endless ... . Relevance: Naturals. Ex: {1,3,5,...}. Depth: Dots indicate.
Finite Set
Limited elements. Relevance: Listable. Ex: {a,e,i,o,u}. Depth: Countable.
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.
- 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.
- Not a set (subjective).
3. Let A = {1, 2, 3, 4, 5, 6}. Is 5 an element of A?
- Yes, 5 ∈ A.
4. Is 8 an element of A where A = {1, 2, 3, 4, 5, 6}?
- No, 8 ∉ A.
5. Is 0 an element of A where A = {1, 2, 3, 4, 5, 6}?
- No, 0 ∉ A.
6. Is 4 an element of A where A = {1, 2, 3, 4, 5, 6}?
- 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).
- {-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, 2, 3, 4, 5}.
9. Write the following set in set-builder form: {3, 6, 9, 12} (key property).
- {x : x is a multiple of 3}.
10. Write the following set in set-builder form: {2,4,8,16,32} (key property).
- {x : x = 2^n, n ∈ N}.
11. List one element of the following set: A = {x : x is an odd natural number}.
- 1 (or any odd natural).
12. List one element of the following set: B = {x : x is an integer, –1/2 < x < 9/2}.
- 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}.
- 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}.
- 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}.
- 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}.
- Matches (iv)-(b).
17. Is the following an example of the null set? {x : x is a natural number, x < 5 and x > 7}.
- Yes, null set.
18. Is the set of even prime numbers a null set?
- No, {2} (only one even prime).
19. Is the set of months of a year finite?
- Yes, finite (12 months).
20. Is the set {1, 2, 3, . . .} infinite?
- 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.
- (i) Yes: Well-defined {January, June, July} (clear criterion).
- (ii) No: Subjective (talent varies).
- Justify: Set if membership definite yes/no.
- Relevance: Well-defined key.
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.
- (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.
- (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.
- (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.
- 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.
- (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.
- (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.
- (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}.
- {-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}.
- {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}.
- {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}.
- {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.
- {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.
- {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}.
- {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}.
- {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}.
- {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, . . .}.
- {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}.
- {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.
- 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].
- (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.
- Step 1: (i) Yes - Clear J months.
- Step 2: (ii) No - Subjective talent.
- Step 3: (iii) No - Best batsmen opinion.
- Step 4: (iv) Yes - Class boys definite.
- Step 5: (v) Yes - N<100 finite.
- Step 6: (vi) Yes - Premchand novels list.
- Step 7: (vii) Yes - Even integers rule.
- Step 8: (viii) Yes - Chapter questions fixed.
- Step 9: (ix) No - Dangerous subjective. Conclusion: Well-defined = set.
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.
- 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.
- (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}.
- (i) {x: x=3n, n∈N}.
- (ii) {x: x=2^n, n∈N}.
- (iii) {x: x=5^n, n∈N}.
- (iv) {x: x even N}.
- (v) {x: x=n^2, 1≤n≤10, n∈N}. Step: Identify pattern.
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}.
- (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].
- 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.
Well-Defined Set
Clear membership. Deriv: Objective. Pitfall: Subjective no. Ex: Odds yes. Interlink: All. Depth: Yes/no test.
Roster Form
List elements. Deriv: Braces. Pitfall: Repeats. Ex: {1,2,3}. Interlink: Finite. Depth: Order no matter.
Set-Builder Form
Property rule. Deriv: : such that. Pitfall: Wrong property. Ex: {x: even}. Interlink: Infinite. Depth: Variable x.
Membership ∈/∉
Belong/not. Deriv: Epsilon. Pitfall: Assume. Ex: 2∈ evens. Interlink: Check. Depth: Basic operation.
Special Sets
N,Z,Q,R. Deriv: Number system. Pitfall: Confuse subsets. Ex: N⊂Z. Interlink: Ch3 functions. Depth: Symbols.
Solution Set
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.
Extended: Empty ∅. Common: Roster infinite dots.
Sets & Representations - Detailed Guide
Well-defined, elements (expanded with table).
| Example Collection | Set? | Reason |
|---|---|---|
| Odd <10 | Yes | Clear {1,3,5,7,9} |
| India rivers | Yes | Listable |
| Best batsmen | No | Subjective |
Tip: Definite = yes. Depth: Objects synonymous elements. Examples: PDF. Graphs: Table. Advanced: Universal.
Principles: Collection. Errors: Vague. Real: Teams.
Extended: ∈ examples.
Roster vs Set-Builder Form - Detailed Guide
Methods compared (expanded with conversions).
Roster
List { }. Finite easy. Ex: {a,e,i,o,u}. Depth: Dots infinite.
Builder
{x: rule}. Rules best. Ex: {x: divisor 42}. Depth: Such that.
Tip: Small list roster; Big rule builder. Depth: No repeats. Examples: Equations. Graphs: Matching table. Advanced: Variable choice.
Principles: Representation. Errors: Order matters no. Real: Menus.
Extended: Infinite handling.
Solved Examples - Book Examples with Simple Explanations
NCERT Examples 1-8 solved step-by-step in simple words.
Example 1: Write the solution set of the equation x² + x – 2 = 0 in roster form.
Simple Explanation: This is finding roots of equation as a set.
- Step 1: Factor equation: (x - 1)(x + 2) = 0.
- Step 2: Roots: x = 1 or x = -2.
- Step 3: Roster: {1, -2}.
- Simple Way: Solve like quadratic, list in {}.
Example 2: Write the set {x : x is a positive integer and x² < 40} in the roster form.
Simple Explanation: Find positives where square less than 40.
- Step 1: √40 ≈ 6.32, so x up to 6.
- Step 2: Check: 1²=1, 2²=4, ..., 6²=36 <40; 7²=49>40.
- Step 3: Roster: {1, 2, 3, 4, 5, 6}.
- Simple Way: Square and compare till exceeds.
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}.
- For infinite: {x : x odd natural}.
- Mnemonic: "Builder = Box with Rule" (BBR - {x : rule}).
- Tip: For patterns/rules, infinite sets.
Special Sets
- N: {1,2,3,...} natural.
- Z: {...,-1,0,1,...} integers.
- Q: Rationals p/q.
- R: All reals.
- Z⁺, Q⁺, R⁺: Positive versions.
- Mnemonic: "Nice Zebras Quickly Run Past Zero" (N Z Q R P Z - positives).
- Tip: N ⊂ Z ⊂ Q ⊂ R.
Empty/Finite/Infinite
- Empty: ∅ or {}, no elements (e.g., natural between 1-2).
- Finite: Countable n(S) natural.
- Infinite: Endless (e.g., primes).
- Mnemonic: "Empty = Nothing; Finite = Few; Infinite = Forever" (ENF).
- Tip: ∅ finite; test contradictions for empty.
Equal Sets
- A = B if same elements (ignore order/repeats).
- Example: {1,2} = {2,1}.
- Mnemonic: "Equal = Exact match, No extras" (EENE).
- Tip: List unique and compare.
Solution Sets & Examples
- Equation roots: x² + x - 2=0 → {1, -2}.
- Vowels: {a,e,i,o,u}.
- Odd <10: {1,3,5,7,9}.
- Mnemonic: "Solve = Set of Roots" (SSR).
- Tip: Factor for quadratics.
Quick Tips for Learning
- Practice: Convert 5 roster to builder daily.
- Common Error: Repeats in roster - avoid!
- Exam Trick: For infinite, use ... ; for empty, check impossible.
- Visual: Draw Venn for equals (overlap full).
Overall Mnemonic: "Sets: Roster Builds Empty Finite Infinite Equals" (RBEFIE - chapter flow). Use flashcards for symbols.
Formulas & Notations - All Key Symbols for Sets
No algebraic formulas, but notations are "formulas" for sets. Full list from chapter with examples.
| Notation | Description | Example | Usage |
|---|---|---|---|
| ∈ | Belongs to (element of set) | 3 ∈ {1,3,5} | Membership test |
| ∉ | Does not belong to | 4 ∉ {1,3,5} | Exclusion |
| { } | Braces for sets/roster | {1,2,3} | List elements |
| {x : property} | Set-builder form | {x : x even} | Rule-based definition |
| : | "Such that" in builder | {x : x > 0} | Condition separator |
| ∅ or {} | Empty set | ∅ | No elements |
| n(S) | Cardinality (number of elements) | n({1,2})=2 | Finite count |
| = | Equal sets (same elements) | {1,2} = {2,1} | Exact match |
| N | Natural numbers | {1,2,3,...} | Positive integers ≥1 |
| Z | Integers | {...,-1,0,1,...} | All whole numbers |
| Q | Rational numbers | {p/q | q≠0} | Fractions |
| R | Real numbers | All on number line | Includes irrationals |
| Z⁺ | Positive integers | {1,2,3,...} | Subset of Z |
| ... | Indicate infinite continuation | {1,3,5,...} | Roster for infinite |
Tip: Memorize symbols: ∈ for in, : for rule. Practice: Write 5 notations daily.
Derivations & Proofs - Solved Step-by-Step
For Sets, "derivations" include solving equations for solution sets and proving equal sets. All from chapter examples.
Derivation 1: Solution Set of x² + x - 2 = 0 (Roster Form)
Step-by-Step Proof:
- Step 1: Rewrite: x² + x - 2 = 0.
- Step 2: Factor: (x + 2)(x - 1) = 0 (sum -product roots).
- Step 3: Set factors =0: x + 2 = 0 → x = -2; x - 1 = 0 → x = 1.
- Step 4: Verify: Plug x=1: 1+1-2=0; x=-2: 4-2-2=0.
- Conclusion: Solution set {1, -2}. Proof: Quadratic formula confirms roots.
Derivation 2: Roster for {x : positive integer, x² < 40}
Step-by-Step Proof:
- Step 1: Condition: x > 0, integer, x² < 40.
- Step 2: Bound: √40 ≈ 6.32, so x ≤ 6.
- Step 3: Test: x=1:1<40; ... x=6:36<40; x=7:49>40.
- Step 4: List: {1,2,3,4,5,6}.
- Conclusion: Finite set. Proof: Inequality strict.
Derivation 3: Set-Builder for {1,4,9,16,25,...}
Step-by-Step Proof:
- Step 1: Pattern: 1=1², 4=2², etc.
- Step 2: General: x = n², n ∈ N.
- Step 3: Verify: For n=1 to 5: Matches list.
- Step 4: Infinite: n unlimited.
- Conclusion: {x : x = n², n ∈ N}. Proof: Square property unique.
Derivation 4: Proof of Equal Sets - Example 8 (i) ALLOY and LOYAL
Step-by-Step Proof:
- Step 1: Roster ALLOY: {A,L,O,Y} (L repeat ignored).
- Step 2: Roster LOYAL: {L,O,Y,A} (L repeat ignored).
- Step 3: Compare: Same elements {A,L,O,Y}.
- Step 4: Definition: Every element of first in second and vice-versa.
- Conclusion: Equal. Proof: Repeats don't count per definition.
Derivation 5: Proof Finite/Infinite - Primes Set
Step-by-Step Proof:
- Step 1: Set {x : x prime, x ∈ N}.
- Step 2: Examples: 2,3,5,7,11,... no end.
- Step 3: Euclid's theorem: Infinite primes (construct new prime > any list).
- Step 4: Cannot list all (endless).
- Conclusion: Infinite. Proof: Mathematical induction on primes.
Tip: For proofs, use definition + verification. Practice: Prove two sets equal.
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