Class 11 Maths · Chapter 1
Short answer:
Set ek well-defined collection of distinct objects hota hai — jaise {1,2,3} ya "saare even numbers ka collection". Class 11 Maths Chapter 1 me Sets ka roster/set-builder representation, types of sets (empty, finite, infinite, equal), subsets, power set, aur operations (union ∪, intersection ∩, difference −, complement ′) cover hote hain. Ye class 11 maths chapter 1 sets ncert solutions page har concept step-by-step working ke saath deta hai.
Class 11 Maths ka Chapter 1 — Sets — poore Class 11 aur Class 12 Maths course ki neev hai. Relations & Functions, Probability, aur yahan tak ki Straight Lines aur Conic Sections jaisi cheezein bhi set-builder notation, intervals, aur union-intersection ki language use karti hain. Isliye jo bhi class 11 maths ncert solutions dhoondh rahe ho, Sets ko theek se samajhna sabse pehla step hai — baaki chapters (jaise class 11 maths chapter 3 trigonometric functions ncert solutions) isi foundation pe khade hote hain.
Ye chapter simple lagta hai kyunki symbols (∈, ⊂, ∪, ∩) roz-marra ki cheezon jaisa lagte hain — par exams me sabse zyada marks students isi chapter me notation confusion (∈ vs ⊂) ki wajah se katate hain. Neeche har concept ke saath step-by-step working di gayi hai, jo class 11 maths chapter 1 sets ncert solutions ke liye CBSE marking-scheme style follow karti hai — final answer nahi, poora process dikhaya gaya hai.
Agar poora class 11 maths syllabus 2026-27 pdf ya class 11 maths ncert book pdf download chahiye toh ncert.nic.in ka official portal hi use karo — yahan diya gaya content us rationalised (14-chapter) syllabus ke exact Chapter 1 ke saath align hai.
Chapter 1 Summary — 5 Minute Revision
Chapter 1 "Sets" ki core cheezein ek jagah:
- Representation: Set ko roster (listing) form ya set-builder (rule-based) form me likha ja sakta hai — dono ek hi set ko denote karte hain.
- Types of sets: Empty set (∅) — koi element nahi; Finite/Infinite — countable ya nahi; Equal sets — same elements, order matter nahi karta.
- Subsets aur Power set: A ⊂ B tab jab A ka har element B me ho. n elements wale set ka power set 2ⁿ subsets rakhta hai, jisme ∅ aur khud A bhi shaamil hote hain.
- Operations: Union (∪) — dono sets ke saare unique elements; Intersection (∩) — sirf common elements; Difference (A−B) — A me hai par B me nahi; Complement (A′) — universal set U ka woh part jo A me nahi.
- Venn diagrams in operations ko visually samajhne ka tool hain — exams me diagram-based questions bhi aate hain.
- Practical/word problems (survey type — cricket/football/tennis, newspaper readers) inclusion-exclusion formula (n(A∪B) = n(A)+n(B)−n(A∩B), aur 3-set version) test karte hain — yahi is chapter ka sabse scoring aur sabse common exam-application part hai.
- Laws: Commutative, Associative, Distributive, De Morgan's, aur Complement laws — proof-based questions me inhi ka use hota hai.
In-Text Questions — Solutions
Q1. Roster form me likho: "word TRIGONOMETRY ke saare distinct letters ka set."
Word likho aur repeat letters hatao: T, R, I, G, O, N, O, M, E, T, R, Y
Set = {T, R, I, G, O, N, M, E, Y}
9 distinct letters hain — kisi bhi set me har element sirf ek baar likha jata hai, repeat nahi.
Q2. A = {2, 4, 6, 8, 10, 12} ko set-builder form me likho.
Pattern dekho — saare even numbers, 2 se 12 tak.
A = {x : x = 2n, n ∈ N, 1 ≤ n ≤ 6}
Q3. Roster form me likho: {x : x ek natural number hai aur x² < 40}.
n = 1,2,3,...try karo:
1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36 (sab < 40), 7² = 49 (> 40, reject)
Set = {1, 2, 3, 4, 5, 6}
Q4. Kya ye ek 'set' hai — "India ke sabse talented 10 writers ka collection"?
Set banne ke liye collection well-defined honi chahiye — matlab kisi bhi object ke liye clearly decide ho sake ki wo collection me hai ya nahi.
"Talented" ek subjective term hai — do alag logon ki list alag ho sakti hai. Isliye yeh set nahi hai.
Q5. Kya A = {x : x ek natural number hai aur x < 1} empty set hai?
Natural numbers ki shuruaat 1 se hoti hai (N = 1,2,3,...). Koi bhi natural number 1 se chota nahi hota.
A = ∅ (empty set), kyunki koi bhi element condition satisfy nahi karta.
Q6. Batao finite hai ya infinite: (i) {x : x ∈ N, x even hai} (ii) {x : x, 99 se chota prime number hai}
(i) Even natural numbers ki koi upper limit nahi — 2,4,6,8,... hamesha chalte rehte hain. Infinite set.
(ii) 99 se chote primes ginti ke hain (2,3,5,...,97) — poori list bana sakte ho. Finite set.
Q7. A = word "FOLLOW" ke distinct letters ka set, B = {F, O, L, W}. Kya A = B?
FOLLOW ke letters: F, O, L, L, O, W — repeats hatao:
A = {F, O, L, W}
B = {F, O, L, W} — dono sets me bilkul same elements hain (order matter nahi karta).
Haan, A = B (equal sets).
Q8. A = {1, 2, {3, 4}, 5}. Check karo: (i) {3,4} ∈ A ? (ii) {3,4} ⊂ A ? (iii) 1 ∈ A ?
(i) {3,4} khud ek element ki tarah A me listed hai, isliye {3,4} ∈ A — True.
(ii) Subset banne ke liye 3 aur 4 individually A ke elements hone chahiye — par A me sirf bundle {3,4} hai, akela 3 ya 4 nahi. Isliye {3,4} ⊂ A — False.
(iii) 1 seedha A me listed hai — True.
Q9. A = {a, b, c} ke saare subsets likho.
Systematically list karo — 0 elements se 3 elements tak:
∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}
Total = 2³ = 8 subsets (∅ aur poora set A dono subsets ginte hain).
Q10. Ek set me 5 elements hain. Uske power set me kitne elements honge?
Power set formula:
n(P(A)) = 2ⁿ = 2⁵ = 32
Q11. Set {x : x ∈ R, −3 < x ≤ 5} ko interval notation me likho.
'<' matlab open bracket, '≤' matlab closed bracket:
Set = (−3, 5]
Q12. Agar A ⊂ B hai, toh A ∪ B kya hoga?
A ⊂ B ka matlab hai A ka har element pehle se hi B me maujood hai. Union lete waqt A se koi naya element nahi judta.
A ∪ B = B
Poore Class 11 Maths ke handwritten colour notes
IITian & district toppers ke banaye short notes — revision-ready, diagram ke saath. Board se pehle poora syllabus 3 din me revise.
Exercise Questions — Solutions (Q1–Q16)
Q1. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, find A ∪ B and A ∩ B.
Step 1: Union me dono sets ke saare elements aate hain, koi repeat nahi hota.
A ∪ B = {1, 2, 3, 4, 5, 6}
Step 2: Intersection me sirf common elements aate hain.
A ∩ B = {3, 4}
Q2. If A = {2, 4, 6, 8, 10}, B = {4, 8, 12, 16}, find A − B and B − A.
A − B = A ke woh elements jo B me nahi hain.
A − B = {2, 6, 10}
B − A = B ke woh elements jo A me nahi hain.
B − A = {12, 16}
Note: A − B ≠ B − A, difference commutative operation nahi hai.
Q3. Universal set U = {1, 2, 3, ..., 10} aur A = {1, 3, 5, 7, 9}. Find A′.
Complement A′ = U − A, yani U ke woh elements jo A me nahi hain.
A′ = {2, 4, 6, 8, 10}
Q4. U = {1,2,...,10}, A = {1,2,3,4}, B = {2,4,6,8}. Verify De Morgan's law (A ∪ B)′ = A′ ∩ B′.
LHS:
A ∪ B = {1, 2, 3, 4, 6, 8}
(A ∪ B)′ = U − (A ∪ B) = {5, 7, 9, 10}
RHS:
A′ = {5, 6, 7, 8, 9, 10}
B′ = {1, 3, 5, 7, 9, 10}
A′ ∩ B′ = {5, 7, 9, 10}
LHS = RHS, hence verified.
Q5. Same sets se, verify (A ∩ B)′ = A′ ∪ B′.
LHS:
A ∩ B = {2, 4}
(A ∩ B)′ = U − {2,4} = {1, 3, 5, 6, 7, 8, 9, 10}
RHS:
A′ ∪ B′ = {5,6,7,8,9,10} ∪ {1,3,5,7,9,10} = {1,3,5,6,7,8,9,10}
LHS = RHS, verified.
Q6. Agar A ⊂ B ho, toh prove karo A ∪ B = B aur A ∩ B = A.
Union: A ⊂ B ka matlab A ka har element pehle se B me hai. Isliye union me A add karne se koi naya element nahi aata.
A ∪ B = B
Intersection: A ka har element B me bhi hai, isliye common elements = poora A.
A ∩ B = A
Q7. Kisi bhi do sets A, B ke liye prove karo A − B = A ∩ B′.
A − B me woh elements hain jo A me hain par B me nahi.
x ∈ (A − B) ⟺ x ∈ A aur x ∉ B ⟺ x ∈ A aur x ∈ B′ ⟺ x ∈ (A ∩ B′)
Dono taraf ke elements same hain, isliye A − B = A ∩ B′.
Q8. Ek class ke 50 students me se 30 cricket khelte hain, 25 football khelte hain, aur 15 dono khelte hain. Kitne students kam se kam ek game khelte hain? Kitne koi game nahi khelte?
Given: n(C) = 30, n(F) = 25, n(C ∩ F) = 15, total = 50
n(C ∪ F) = n(C) + n(F) − n(C ∩ F) = 30 + 25 − 15 = 40
Kam se kam ek game khelne wale = 40 students
Koi game nahi khelne wale = 50 − 40 = 10
Q9. 100 students ke survey me 60 Hindi newspaper padhte hain, 40 English padhte hain, 20 dono padhte hain. Sirf Hindi, sirf English, aur koi bhi nahi padhne walon ki ginti nikalo.
Sirf Hindi = n(H) − n(H∩E) = 60 − 20 = 40
Sirf English = n(E) − n(H∩E) = 40 − 20 = 20
Kam se kam ek = 60 + 40 − 20 = 80
Koi bhi nahi = 100 − 80 = 20 students
Q10. 60 logon ke group me 25 chai pasand karte hain, 20 coffee pasand karte hain, 15 dono pasand karte hain. Baaki kuch bhi pasand nahi karte. Kitne log kuch bhi pasand nahi karte?
n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 25 + 20 − 15 = 30
Neither = 60 − 30 = 30 log
Q11. 25 students ke group me 12 cricket, 11 football, 10 tennis khelte hain. 5 cricket-football dono, 4 football-tennis dono, 3 cricket-tennis dono, aur 2 teeno khelte hain. Kitne students kam se kam ek game khelte hain? Kitne koi game nahi khelte?
3-set formula (inclusion-exclusion):
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
= 12 + 11 + 10 − 5 − 4 − 3 + 2 = 23
Kam se kam ek game = 23 students
Koi game nahi = 25 − 23 = 2 students
Q12. A = {x : x, 3 ka multiple hai, x ≤ 30}, B = {x : x, 5 ka multiple hai, x ≤ 30}. Find A ∩ B aur A ∪ B.
A = {3,6,9,12,15,18,21,24,27,30} — 10 elements
B = {5,10,15,20,25,30} — 6 elements
A ∩ B = jo 3 aur 5 dono ke multiple hain, yani 15 ke multiple:
A ∩ B = {15, 30}
A ∪ B = {3,5,6,9,10,12,15,18,20,21,24,25,27,30}
Q13. Prove karo: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) — distributive law.
Let x ∈ A ∪ (B ∩ C).
⟹ x ∈ A, ya x ∈ (B ∩ C) ⟹ x ∈ A, ya (x ∈ B aur x ∈ C)
Dono cases me x ∈ (A∪B) aur x ∈ (A∪C) satisfy hota hai, isliye x ∈ (A∪B) ∩ (A∪C).
Reverse direction bhi similarly hold karta hai (dono taraf se contain), isliye:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Q14. Prove karo: A − (B ∪ C) = (A − B) ∩ (A − C).
Let x ∈ A − (B ∪ C) ⟹ x ∈ A aur x ∉ (B ∪ C) ⟹ x ∈ A aur (x ∉ B aur x ∉ C)
⟹ (x ∈ A aur x ∉ B) aur (x ∈ A aur x ∉ C) ⟹ x ∈ (A−B) ∩ (A−C)
Yeh step reversible hai, isliye equality hold karti hai:
A − (B ∪ C) = (A − B) ∩ (A − C)
Q15. A = {1,2,3}, B = {2,3,4}. Numerically verify n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
A ∪ B = {1,2,3,4} ⟹ n(A∪B) = 4
n(A) = 3, n(B) = 3, A ∩ B = {2,3} ⟹ n(A∩B) = 2
n(A) + n(B) − n(A∩B) = 3 + 3 − 2 = 4
LHS = RHS = 4, verified.
Q16. n(A) = 20, n(B) = 28, n(A ∪ B) = 36. Find n(A ∩ B).
Formula rearrange karo:
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
n(A ∩ B) = 20 + 28 − 36 = 12
Important Equations — Ek Nazar Me
| Concept | Formula / Rule |
|---|---|
| Power set size | n(P(A)) = 2ⁿ, jahan n = n(A) |
| Proper subsets count | Proper subsets = 2ⁿ − 1 |
| Union of two sets (count) | n(A ∪ B) = n(A) + n(B) − n(A ∩ B) |
| Union of three sets (count) | n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C) |
| Difference in terms of complement | A − B = A ∩ B′ |
| De Morgan's Law 1 | (A ∪ B)′ = A′ ∩ B′ |
| De Morgan's Law 2 | (A ∩ B)′ = A′ ∪ B′ |
| Commutative laws | A ∪ B = B ∪ A, A ∩ B = B ∩ A |
| Associative laws | (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| Distributive laws | A ∪ (B ∩ C) = (A∪B) ∩ (A∪C); A ∩ (B ∪ C) = (A∩B) ∪ (A∩C) |
| Identity laws | A ∪ ∅ = A, A ∩ U = A |
| Complement laws | A ∪ A′ = U, A ∩ A′ = ∅, (A′)′ = A |
| Subset relation shortcut | Agar A ⊂ B, toh A ∪ B = B aur A ∩ B = A |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- ∈ aur ⊂ mix karna: 3 ∈ {1,2,3} sahi hai, par {3} ∈ {1,2,3} galat hai — {3} ⊂ {1,2,3} likhna chahiye. Element aur subset alag cheezein hain.
- n(A∪B) formula me −n(A∩B) bhool jaana: Students n(A)+n(B) hi likh dete hain, common elements do baar count ho jaate hain — survey/word-problems me sabse common galti.
- Power set count galat karna: 2ⁿ me ∅ aur poora set A bhi shaamil hote hain — inhe 'chhota' ya 'trivial' samajh kar exclude kar dena galat hai, jawab 2ⁿ−2 aa jaata hai jo wrong hai.
- A−B aur B−A ko same maan lena: Set difference commutative nahi hai — A−B ke elements sirf A me hote hain, B−A ke sirf B me. Dono alag sets ho sakti hain.
- De Morgan's laws swap kar dena: (A∪B)′ = A′∩B′ aur (A∩B)′ = A′∪B′ — students interchange kar dete hain. Yaad rakhne ka tareeka: complement lete waqt ∪ aur ∩ aapas me switch hote hain.
- Roster form me repeat elements likhna: Set me har element unique hota hai — {1,1,2,3} likhna galat hai, sahi {1,2,3} hai. Same galti word-se-set banate waqt (jaise letters of a word) bhi hoti hai.
Board-Style Important Questions
- 1 mark: Set A = {a, b} ka power set likho.
- 2 marks: A = {1,2,3,4,5}, B = {2,4,6,8} diya hai. A ∩ B aur A ∪ B dono find karo.
- 3 marks: U = {1,...,10}, A = {1,2,3,4}, B = {2,4,6,8}. De Morgan's law (A∪B)′ = A′∩B′ ko verify karo, poora step dikhao.
- 4 marks: 50 students ki class me 30 cricket khelte hain, 25 football khelte hain, 15 dono khelte hain. Kam se kam ek game khelne walon ki aur kisi ka bhi game na khelne walon ki ginti nikalo.
- 5 marks: 25 students ke group me 12 cricket, 11 football, 10 tennis khelte hain; 5 cricket-football dono, 4 football-tennis dono, 3 cricket-tennis dono, aur 2 teeno khelte hain. Poora inclusion-exclusion working dikhate hue batao kitne students kam se kam ek game khelte hain.
Aksar Poochhe Jaane Wale Sawaal
Class 11 Maths Chapter 1 Sets ke important questions with solutions kahan practice karein?
Is page ke intext aur exercise sections dono me step-by-step solved questions diye hain jo union, intersection, complement, aur word-problems (survey type) cover karte hain — yeh class 11 maths important questions with solutions ka core set hai. Board-level practice ke liye NCERT ke apne exercises (1.1 se Miscellaneous tak) already isi pattern pe based hain.
Sets chapter ke baad, class 11 maths syllabus 2026-27 me kaunsa chapter aata hai?
Rationalised syllabus me Sets (Ch 1) ke baad Relations and Functions (Ch 2) hai, phir Trigonometric Functions (Ch 3) — jiske liye alag se class 11 maths chapter 3 trigonometric functions ncert solutions page dekh sakte ho. Poori sequence ke liye official class 11 maths syllabus 2026-27 pdf ncert.nic.in se download karo.
Poore chapter ke formulas ek jagah kahan milenge — class 11 maths formulas pdf download kaise karein?
Is page ke 'formulas' section me Sets ka poora formula table hai (union-intersection count, De Morgan's laws, power set count). Poore syllabus ka combined class 11 maths formulas pdf download chahiye toh official class 11 maths ncert book pdf download ke saath NCERT ka end-of-book summary bhi check karo.
Class 11 Maths exemplar solutions, NCERT textbook solutions se kaise different hote hain?
NCERT textbook questions foundation build karte hain, jabki class 11 maths exemplar solutions usi topic pe HOTS (higher-order thinking) level ke tougher, multi-step problems dete hain — Sets chapter me typically proof-based aur multi-set word problems zyada aate hain exemplar me.
Sets chapter aage kis-kis chapter me directly use hota hai?
Sets ka interval-notation aur set-builder form seedha class 11 maths straight lines ncert solutions (domain/region conditions), class 11 maths limits and derivatives ncert solutions (domain restrictions), class 11 maths probability important questions (sample space set hi hota hai) aur class 11 maths conic sections notes me use hota hai. Isliye Sets skip karna aage risky hai.
Class 11 Maths chapter wise weightage 2026-27 me Sets ka kitna weight hai?
Alag-alag sources me is par conflicting numbers milte hain, isliye yahan koi specific marks-weightage figure invent nahi kiya ja raha. Accurate class 11 maths chapter wise weightage 2026-27 ke liye CBSE ka official curriculum document hi check karo — wahi authoritative source hai.
Class 11 Maths — Saare Chapters
Board exam tak sirf revision karna hai?
Class 11 Maths ke saare chapters ke colour handwritten short notes — diagrams, formulas aur important points ek jagah.
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