Sets

Sets

Maths ka sabse basic building block hai set. Aage jo bhi padhoge — relations, functions, geometry, probability — sab sets ke upar khade hain. Is chapter me samjhenge ki set hota kya hai, use likhne ke tareeke, alag-alag types, aur sets par operations.

Ek set well-defined collection hoti hai distinct (alag-alag) objects ki. "Well-defined" ka matlab — kisi bhi object ke liye hum saaf-saaf decide kar sakein ki wo us collection me hai ya nahi.

Jaise "India ki rivers" ek set hai, lekin "India ke 10 sabse talented writers" set nahi hai — kyunki "talented" har insaan ki opinion par depend karta hai, wo well-defined nahi.

1. Basic Notation

• Sets ko capital letters se likhte hain: \(A, B, C, X, Y, \dots\)
• Set ke objects uske elements (members) kehlate hain, small letters me: \(a, b, c, x, \dots\)
• Agar \(a\), set \(A\) ka element hai to likhte hain \(a \in A\) (padho: "\(a\) belongs to \(A\)").
• Agar \(b\) element nahi hai to \(b \notin A\).

Toh agar \(V\) vowels ka set hai, to \(a \in V\) par \(b \notin V\). Kuch standard sets ke fixed symbols hote hain:

2. Set Likhne ke Do Tareeke

Roster (tabular) form

Saare elements braces ke andar, commas se separate karke likho. Jaise 7 se chote even positive integers ka set \(\{2, 4, 6\}\).

• Order matter nahi karta: \(\{1, 3, 7, 21\}\) aur \(\{21, 1, 7, 3\}\) same set hain.
• Elements repeat nahi hote: SCHOOL word ke letters ka set \(\{S, C, H, O, L\}\).
• Teen dots ("\(\dots\)") batate hain ki pattern aage chalta rehta hai, jaise odd naturals \(= \{1, 3, 5, \dots\}\).

Set-builder form

Yahan elements ki common property likhte hain. Jaise,

\[ V = \{\, x : x \text{ is a vowel in the English alphabet} \,\} \]

Padho: "saare \(x\) ka set aise ki \(x\) ek vowel hai." Yahan colon ":" ka matlab "such that" (aise ki). Ek aur example:

\[ A = \{\, x : x \in \mathbb{N},\ 3 < x < 10 \,\} = \{4, 5, 6, 7, 8, 9\} \]

3. Empty Set

Jis set me koi element nahi hota, use empty set (null/void set) kehte hain, symbol \(\varnothing\) ya \(\{\,\}\).

Empty sets ke examples:

• \(\{\, x : x \in \mathbb{N},\ 1 < x < 2 \,\}\) — 1 aur 2 ke beech koi natural number nahi.
• \(\{\, x : x^{2} - 2 = 0,\ x \in \mathbb{Q} \,\}\) — \(\sqrt{2}\) rational nahi hai.
• \(\{\, x : x \text{ is an even prime} > 2 \,\}\) — 2 hi ek matra even prime hai.

4. Finite aur Infinite Sets

Set \(S\) ke distinct elements ki ginti ko \(n(S)\) likhte hain.

Set finite hai agar wo empty ho ya uske elements ki definite ginti ho; warna infinite.

\(A = \{1,2,3,4,5\}\) ke liye \(n(A) = 5\), toh finite. Natural numbers \(\mathbb{N}\) infinite hai. Infinite sets ko dots ke saath likhte hain, jaise \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\).

5. Equal Sets

Do sets \(A\) aur \(B\) equal (\(A = B\)) tab hote hain jab dono me bilkul same elements hon.

Jaise \(\{1,2,3\} = \{3,1,2\}\) (order matter nahi), par \(\{1,2,3\} \neq \{1,2,3,4\}\).

6. Subsets

\(A\), \(B\) ka subset hai (\(A \subseteq B\)) agar \(A\) ka har element \(B\) me bhi ho. Symbol me: \(a \in A \Rightarrow a \in B\).

• Har set apna khud ka subset hota hai: \(A \subseteq A\).
• Empty set har set ka subset hai: \(\varnothing \subseteq A\).
• Agar \(A \subseteq B\) par \(A \neq B\), to \(A\) proper subset hai: \(A \subset B\).
• \(A = B\) tabhi jab \(A \subseteq B\) aur \(B \subseteq A\) dono ho.

Intervals (\(\mathbb{R}\) ke subsets)

Real numbers \(a < b\) ke liye:

• Open interval: \((a, b) = \{\, x : a < x < b \,\}\) — endpoints chhute nahi.
• Closed interval: \([a, b] = \{\, x : a \le x \le b \,\}\) — endpoints included.
• Half-open: \([a, b)\) aur \((a, b]\).

7. Power Set

\(A\) ka power set \(P(A)\) hota hai — \(A\) ke saare subsets ka set.

Agar \(A\) me \(n\) elements hain, to \(P(A)\) me \(2^{n}\) elements hote hain. Jaise \(A = \{1, 2\}\) ho to

\[ P(A) = \{\, \varnothing,\ \{1\},\ \{2\},\ \{1,2\} \,\}, \qquad n\big(P(A)\big) = 2^{2} = 4. \]

8. Universal Set

Kisi bhi discussion me hum ek bada set fix karte hain jisme saare under-consideration sets aa jaate hain — ise universal set \(U\) kehte hain. Jaise integers ke sets padhte waqt \(U = \mathbb{Z}\) le sakte hain.

9. Venn Diagrams

Venn diagram me universal set \(U\) ko rectangle se aur baaki sets ko uske andar circles se dikhate hain. Isse sets ke relations aur operations samajhna aasaan ho jaata hai.

10. Sets par Operations

Union

\(A \cup B = \{\, x : x \in A \text{ or } x \in B \,\}\) — dono me se kisi ek me bhi ho.

Intersection

\(A \cap B = \{\, x : x \in A \text{ and } x \in B \,\}\) — jo dono me common ho.

Agar \(A \cap B = \varnothing\), to dono me kuch common nahi — inhe disjoint kehte hain.

Difference

\(A - B = \{\, x : x \in A \text{ and } x \notin B \,\}\) — jo \(A\) me ho par \(B\) me nahi.

Complement

\(A\) ka complement (\(U\) ke respect me) \(A' = U - A = \{\, x : x \in U,\ x \notin A \,\}\).

11. Set Operations ke Laws

Kisi bhi sets \(A, B, C\) ke liye (universal set \(U\) ke andar):

• Idempotent: \(A \cup A = A\), \(\quad A \cap A = A\).
• Identity: \(A \cup \varnothing = A\), \(\quad A \cap U = A\).
• Commutative: \(A \cup B = B \cup A\), \(\quad A \cap B = B \cap A\).
• Associative: \((A \cup B) \cup C = A \cup (B \cup C)\).
• Distributive: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).
• De Morgan's laws: \((A \cup B)' = A' \cap B'\) aur \((A \cap B)' = A' \cup B'\).

12. Counting Formula

Do finite sets ke union me elements ki ginti:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B). \]

\(n(A \cap B)\) isliye minus karte hain taaki common elements do baar count na hon. Agar \(A\) aur \(B\) disjoint hain to \(n(A \cup B) = n(A) + n(B)\).

Key Takeaways

• Set = distinct objects ki well-defined collection; roster ya set-builder form me likho.
• \(\varnothing\) empty set hai; \(n(S)\) se set finite ya infinite decide hota hai.
• \(A \subseteq B\) matlab \(A\) ka har element \(B\) me; power set \(P(A)\) me \(2^{n}\) members.
• Main operations: union \(\cup\), intersection \(\cap\), difference \(-\), complement \('\).
• De Morgan's laws aur \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\) yaad rakho.