Lesson 2 — Beginner

Set Theory & Operations

Set theory discrete maths ki neev hai — logic, relations, probability sab isi par khade hote hain. Is lesson mein seekhenge set kya hota hai, uske operations (union, intersection, difference), subsets, power set aur De Morgan laws — sab worked examples ke saath.

⏱ 14 min• Beginner• Introduction

Set kya hai

WHAT

Set ek well-defined collection of distinct objects hai. Objects ko elements kehte hain. A = {1, 2, 3} — ye teeno A ke elements hain. Order matter nahi karta, repeat nahi hota ({1,1,2} bhi {1,2} hi hai).

WHEN

Jab bhi group of cheezon se deal karo — class ke students, pass marks, selected candidates — har jagah set ban jaata hai. Database queries, probability, logic sab set theory par chalti hain.

WHERE

Maths, programming (arrays aur HashSet), database (SQL ka SELECT result bhi set hi hai) aur exams — har jagah. GATE ka sabse pehla topic yahi hai.

Set ka size (cardinality) |A| se likha jaata hai — |{1,2,3}| = 3. Element ka set mein hona ∈ symbol se likhte hain, na hona ∉ se.

Set ki representations

Roster (listing) method : A = {1, 2, 3, 4, 5}
Set-builder (rule)      : A = {x : x is natural, x < 6}
                           (x aisa hai jo natural ho, 6 se chhota)

Special sets :
N = {1, 2, 3, ...}        natural numbers
Z = {..., -1, 0, 1, ...}  integers
Q = p/q form rationals,  R = real numbers
Empty set  = {} = ∅        size 0
Universal set U : sabhi sets ka bada group

Notation yaad rakho: 2 ∈ A ka matlab hai 2 set A mein hai. 7 ∉ A ka matlab 7 A mein nahi hai. B ⊆ A ka matlab B ka har element A mein hai (subset). Ye teeno symbols exam mein aksar aate hain.

Set operations (union, intersection, difference)

Do sets ke saath 4 kaam sabse zyada use hote hain — union, intersection, difference aur complement:

Let A = {1, 2, 3, 4},  B = {3, 4, 5, 6}

A ∪ B (union)        = {1, 2, 3, 4, 5, 6}   dono ke elements
A ∩ B (intersection) = {3, 4}               sirf common elements
A − B (difference)   = {1, 2}                A mein jo B mein nahi
A′ (complement)      = U − A                 universal set se bacha

Cardinality formula (inclusion-exclusion):
|A ∪ B| = |A| + |B| − |A ∩ B|
        = 4 + 4 − 2 = 6

Ye formula bar bar aata hai — do baar count ho gaye elements (intersection) ko ek baar minus karna padta hai. 3 sets ke liye bhi wahi logic hai, bas aur terms add hoti hain.

Subsets aur power set

Subset ka matlab andar ka andar. Har set apna khud ka subset hota hai, aur ∅ har set ka subset hota hai:

A ⊆ B : har element of A, element of B bhi hai
A ⊂ B : proper subset (A ≠ B, matlab A B mein strict chhota)

Number of subsets of n-element set = 2^n
Power set P(A) = sab subsets ka set, |P(A)| = 2^n

Example: A = {a, b}
P(A) = {∅, {a}, {b}, {a, b}}  → size 4 = 2^2

Cartesian product:
A × B = {(x, y) : x ∈ A, y ∈ B}, |A × B| = |A| × |B|

Interview question: Agar |A| = 3 hai toh A ke kitne subsets hain? Answer 2^3 = 8. Aur |P(A)| = 8. Ye formula GATE mein count karne wale questions ka foundation hai.

Set laws aur De Morgan

Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A.
Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C) — bracket ki parwah nahi.
Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Identity: A ∪ ∅ = A, A ∩ U = A.
Complement: (A′)′ = A, A ∪ A′ = U, A ∩ A′ = ∅.

De Morgan ke laws (sabse zyada poochhe jaate hain):
(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′

Yaad rakho: union ka complement → complements ka intersection
Aur intersection ka complement → complements ka union

Worked: A = {1,2,3}, B = {3,4}, U = {1,2,3,4,5}
A ∪ B = {1,2,3,4} → (A∪B)′ = {5}
A′ = {4,5}, B′ = {1,2,5} → A′∩B′ = {5} ✓

Exercise

Question: A={1,2,3}, B={2,3,4}. A ∩ B mein kaunse elements hain? (comma se, chhota se bada order mein)

Question: |A|=5, |B|=3, aur |A∩B|=2 hai. |A∪B| kitna hai? Sirf number daalo.

Question: 2 elements wale set ke subsets kitne hote hain? Sirf number daalo.

Common mistakes

{2} aur 2 mein confusion: 2 ∈ A element hai, {2} ⊆ A subset hai. Exam mein dono tarah ke options rakhe jaate hain — symbol dhyan se padho.
Union aur intersection ulta: Union = sab kuch jodna (badha), intersection = sirf common (chhota). ∪ dekh kar jodna, ∩ dekh kar common yaad rakho.
Inclusion-exclusion mein second half chhod dena: |A∪B| = |A|+|B|−|A∩B| — intersection minus karna zaroori hai, warna common elements do baar count ho jaate hain.

Desi example: Canteen ka chai-coffee survey

Canteen mein 60 students the. Survey hua: 35 chai pite hain, 30 coffee, aur 20 dono. Ab canteen waala poochhta hai — kitne log toh kuch bhi nahi pite? Inclusion-exclusion se:

Chai = 35, Coffee = 30, Dono = 20
Chai ya Coffee = 35 + 30 − 20 = 45
Koi nahi pita  = 60 − 45 = 15 students
Formula: |A ∪ B| = |A| + |B| − |A ∩ B|

Exam mapping (GATE / CUET)

GATE: inclusion-exclusion aur power set size — almost har saal 1-2 marks.
CUET/B.Sc: set-builder vs roster representation direct question.
Trap: "|A∪B| = |A|+|B|" — galat jab common elements ho, minus karna padta hai.
Trap: "∅ ka size 1 hai" — galat, empty set ka size 0 hai.

Lesson complete?

Sets ka operations clear. Agla lesson — Propositional Logic — mein dekhenge sahi aur galat arguments kaise check karte hain (truth tables ke saath).