Lesson 4 — Beginner

Predicates & Quantifiers (∀, ∃)

Propositional logic mein statements fixed the — ab variables aa jaate hain. "x > 5" ek proposition nahi hai (x ke bina true/false nahi), isliye predicate aur quantifier chahiye. Is lesson mein seekhenge ∀ aur ∃ kaise kaam karte hain aur inki negation kaise hoti hai.

⏱ 14 min• Beginner• Propositional Logic

Predicate kya hai

WHAT

Predicate ek open statement hai jisme variable ho — jaise P(x): "x > 5". Jab x ki value daalo, statement true ya false ho jaati hai. Variable ke saath domain (allowed values) deni padti hai.

WHEN

Jab statement mein har (all) ya koi na koi (some) aaye — tab quantifier chahiye. "Sab students pass hue" — ye predicate + universal quantifier hai.

WHERE

Database queries (WHERE clauses), mathematical proofs (for all n...), aur GATE ke logic questions — sab predicates par chalte hain.

Predicate ko P(x), Q(x, y) jaisa likhte hain — P function jaisa hai jo true/false deta hai. Isko poori statement banane ke liye quantifier ( ∀ ya ∃ ) lagana padta hai.

Universal (∀) aur Existential (∃) quantifier

Universal quantifier  ∀x : "har x ke liye" (for all)
Existential quantifier ∃x : "koi ek x" (there exists)

Domain: integers
∀x (x + 1 > x)  → TRUE   (har integer ke liye satya)
∃x (x² = 4)      → TRUE   (x = 2 ya x = −2 hai)
∀x (x > 5)       → FALSE  (x = 0 hi counterexample hai)
∃x (x < 0)       → TRUE   (x = −1)
∀x (x is even)   → FALSE  (odd numbers exist karte hain)

Trick: ∀ ka matlab hai ek bhi counterexample nahi hona chahiye. ∃ ka matlab hai ek example mil gayi — kaam ho gaya. Prove karna ho toh ∀ ke liye general value rakho, ∃ ke liye ek value dhoond ke dikha do.

Quantifier ki negation

Quantifier ki negation mein ek hi cheez badalti hai — ∀ becomes ∃, aur ∃ becomes ∀ — saath mein andar wala statement ulta ho jaata hai:

¬(∀x P(x))  ≡  ∃x ¬P(x)    "har x true" ka ulat = "koi x false"
¬(∃x P(x))  ≡  ∀x ¬P(x)    "koi x true" ka ulat = "har x false"

Example 1:
Statement : "Sab students cricket khelte hain"  = ∀x P(x)
Negation  : "Koi student cricket nahi khelta"   = ∃x ¬P(x)
(ek bhi student jo na khelta ho — statement gir jaayegi)

Example 2:
Statement : "Class mein koi na koi topper hai"  = ∃x T(x)
Negation  : "Class mein koi topper hi nahi"     = ∀x ¬T(x)

Yaad rakho: negation mein dono cheezein badalti hain — quantifier bhi aur predicate bhi. Sirf ek badalna sabse common galti hai.

Multiple quantifiers — order matters

Do quantifiers ke order ka matlab badal jaata hai — ye GATE ka favourite trap hai. Hamesha sequence mein padho:

Domain: integers

∀x ∃y (x + y = 0)   → TRUE
  (har x ke liye y = −x mil jaata hai — jaise x=7, y=−7)

∃y ∀x (x + y = 0)   → FALSE
  (ek FIXED y chahiye jo HAR x ke saath jama ho —
   y = −5 lo toh x = 7 pe: 7 − 5 = 2 ≠ 0, fail!)

Isliye: ∀x∃y ≠ ∃y∀x — order ka matlab hi badal jaata hai

Exam trap: Multiple quantifiers wale question mein pehle order padho — pehla quantifier andar wale ko decide karta hai. Ek counterexample dhoond lo, answer turant mil jaayega. ∃y∀x wali statement HAMESHA zyada strong hoti hai.

Exercise

Question: "Koi student cricket khelta hai" — isme kaunsa quantifier lagega? (∃ ya ∀)

Question: ∀x P(x) ka negation kya hai? (∃x ¬P(x) / ∀x ¬P(x) / ∃x P(x))

Question: Domain {1,2,3} hai, P(x): x even hai. ∃x P(x) true hai ya false? (true / false)

Common mistakes

Negation mein sirf statement ulta karna: Quantifier bhi badalta hai — ∀ ka negation ∃ hai, ∃ ka ∀. Sirf P(x) ko ¬P(x) bana dena insufficient hai.
Domain ignore karna: ∀x (x > 10) — domain se farak padta hai. Integers mein false, positives-only mein true. Domain hamesha check karo.
∀∃ aur ∃∀ ko same samajhna: ∀x∃y aur ∃y∀x alag statements hain. ∃y∀x wali zyada strong hoti hai — uske true hone ki bahut kam ummeed hoti hai.

Desi example: Hostel ki light ka rule

Warden ne notice lagaya: "Har room mein kam se kam ek bulb jalna chahiye" — ye ∀x P(x) hai. Ab ek shikayat aayi: "Koi room toh aisa hai jo andhera hai" — ye ∃x ¬P(x) — aur dono ek doosre ko directly maarte hain:

Rule   : "Har room mein bulb hai"  = ∀x P(x)
Ulat  : "Koi room bina bulb hai"   = ∃x ¬P(x)
Ek bhi room agar andhera → rule toot gaya (guaranteed)
Slogan: ∀ ka ulat ∃, ∃ ka ulat ∀ — dono hamesha badlo

Exam mapping (GATE / CUET)

GATE: quantifier ki negation — 1 mark ka direct question.
GATE: ∀x∃y vs ∃y∀x wale trap questions aksar aate hain.
CUET: "meaning of ∀" jaise basic definitions.
Trap: "¬∀xP(x) = ∀x¬P(x)" — galat, ∃x¬P(x) hai.

Lesson complete?

Quantifiers clear. Agla lesson — Mathematical Induction — jo proof karne ka sabse classical tool hai (summation aur divisibility formulas ke liye).