Sets aur Relations | AIkiPadhai

🔵 Sets — Definition aur Representation

Set ek well-defined collection of objects hai. "Well-defined" matlab: clearly pata ho koi cheez set mein hai ya nahi.

Roster Form: Elements list karte hain — A = {1, 2, 3, 4, 5} Set Builder Form: Rule define karte hain — A = {x : x ∈ N, x ≤ 5}

Types of Sets

Type	Definition	Example
Empty Set (∅)	Koi bhi element nahi	{} ya ∅
Singleton Set	Exactly 1 element	{5}
Finite Set	Count ho sakta hai	{1,2,3,4,5}
Infinite Set	Count nahi hota	N = {1,2,3,...}
Universal Set (U)	Sabka superset	Problem ke hisab se
Power Set P(A)	All subsets	If A={1,2}, P(A)={∅,{1},{2},{1,2}}

Agar A mein n elements hain → Subsets = 2ⁿ, Power set mein 2ⁿ elements

Subsets aur Intervals

A ⊆ B: Har element of A, B mein bhi hai Proper subset A ⊂ B: A ⊆ B but A ≠ B Intervals (Real Numbers): [a,b] = {x : a ≤ x ≤ b} (closed) (a,b) = {x : a < x < b} (open) [a,b) = {x : a ≤ x < b} (half-open)

Set Operations

A ∪ B = Elements in A ya B ya dono mein A ∩ B = Elements jo A aur B dono mein hain A − B = Elements jo A mein hain but B mein nahi A' = Elements jo U mein hain but A mein nahi De Morgan's Laws: (A ∪ B)' = A' ∩ B' (A ∩ B)' = A' ∪ B'

Practical Problem — Venn Diagram

Example: 100 students — 60 maths padhte hain, 50 science padhte hain, 30 dono padhte hain. Sirf ek padhne wale kitne?

n(M) = 60, n(S) = 50, n(M∩S) = 30
n(M∪S) = 60 + 50 − 30 = 80 (at least one padhte hain)
Sirf Maths: 60 − 30 = 30
Sirf Science: 50 − 30 = 20
Dono: 30
Koi nahi: 100 − 80 = 20

↔️ Relations

Relation ek set ke do elements ke beech ek mapping hai — actually Cartesian product ka subset hai.

Cartesian Product: A × B = {(a,b) : a ∈ A, b ∈ B} If |A| = m, |B| = n → |A × B| = mn Relation R: A → B means R ⊆ A × B Domain of R = set of first elements Range of R = set of second elements

Example: A = {1,2}, B = {a,b,c}

A × B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)} — 6 elements
R = "first element + 1 = 2" → R = {(1,a)} ← ek possible relation
R = {(1,a),(2,b)} → Domain = {1,2}, Range = {a,b}