16 Set theory
📘 Set Theory – Basic to Advanced
(UGC NET Economics: Complete Concepts + ALL Exam Questions & Solutions)
यह संकलन बेसिक से एडवांस कॉन्सेप्ट और UGC NET Economics (Mathematical Economics) में अब तक पूछे गए सभी Set Theory प्रश्नों को पूर्ण सॉल्यूशन के साथ प्रस्तुत करता है।
⭐ कोई भी शब्द / प्रश्न / उत्तर नहीं हटाया गया है — पहले दिए गए सभी 15+ प्रश्न + UGC NET Official Questions यहाँ सम्मिलित हैं।
🟢 Level 1: Basic Concepts
1. Set क्या है?
Set अच्छी तरह से परिभाषित वस्तुओं (objects) का एक संग्रह है।
उदाहरण:
A = {1, 2, 3, 4, 5}
B = {x : x एक सम संख्या है, x < 10} = {2, 4, 6, 8}
2. Set के प्रकार
| Set Type | Definition | Example |
|---|---|---|
| Finite Set | सीमित संख्या में elements | {1, 2, 3} |
| Infinite Set | असीमित संख्या में elements | {1, 2, 3, …} |
| Null Set (∅) | कोई element नहीं | {} |
| Universal Set (U) | सभी possible elements | U = {सभी पूर्णांक} |
| Subset (⊆) | A के सभी elements B में हों | {1, 2} ⊆ {1, 2, 3} |
| Proper Subset (⊂) | A ⊂ B, लेकिन A ≠ B | {1, 2} ⊂ {1, 2, 3} |
| Power Set (P(A)) | A के सभी subsets का set | A={1,2} → P(A)={∅,{1},{2},{1,2}} |
Cardinality (|A|): Set में elements की संख्या। उदाहरण: |{a, b, c}| = 3
🟡 Level 2: Set Operations
3. Basic Operations
| Operation | Symbol | Definition | Example (A={1,2,3}, B={2,3,4}) |
|---|---|---|---|
| Union | A ∪ B | A या B में सभी elements | {1,2,3,4} |
| Intersection | A ∩ B | A और B दोनों में common | {2,3} |
| Difference | A – B | A में लेकिन B में नहीं | {1} |
| Complement | A' या A^c | U में लेकिन A में नहीं | U={1,2,3,4,5} → A'={4,5} |
| Symmetric Difference | A Δ B | (A-B) ∪ (B-A) | {1,4} |
Disjoint Sets: A ∩ B = ∅, उदा. A = {1,2}, B = {3,4}
🟠 Level 3: Important Laws of Set Theory
Identity Laws: A ∪ ∅ = A | A ∩ U = A
Complement Laws: A ∪ A' = U, A ∩ A' = ∅, (A')' = A, ∅' = U, U' = ∅
Idempotent Laws: A ∪ A = A, A ∩ A = A
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)
De Morgan's Laws (बहुत महत्वपूर्ण): (A ∪ B)' = A' ∩ B' और (A ∩ B)' = A' ∪ B'
🔴 Level 4: Advanced Concepts (UGC NET Level)
Relations
Ordered Pair (a,b), Cartesian Product A×B, Equivalence Relation (Reflexive+Symmetric+Transitive).
Functions (Mapping)
Injective (One-One): a≠b → f(a)≠f(b) | Surjective (Onto) | Bijective = Injective+Surjective.
Convex Set (Economics Application)
सेट S convex है यदि किन्हीं दो बिंदुओं x,y∈S और λ∈[0,1] के लिए λx + (1-λ)y ∈ S हो।
उदाहरण: Budget set { (x,y) : p₁x + p₂y ≤ M } convex होता है।
Countability
Countably Finite, Countably Infinite (N, Q), Uncountable (R, (0,1))
📊 UGC NET Economics में Set Theory के Applications
| Economics Topic | Set Theory Concept Used |
|---|---|
| Consumer Choice | Budget set, Choice set |
| Production Possibility Frontier | Convex set |
| Indifference Curves | Upper contour set, convex preferences |
| Game Theory | Strategy sets, payoff sets |
| Social Choice | Preference aggregation, Arrow's theorem |
| General Equilibrium | Feasible set, Pareto set |
📝 सभी UGC NET Economics प्रश्न और उनके पूर्ण सॉल्यूशन (शुरू से अंत तक)
नीचे वे सभी प्रश्न दिए गए हैं जो पहले दिए गए थे – Basic Level, UGC NET Official Questions, Matching Type, Laws of Sets, Convex Sets, Cardinality, Subset, Disjoint, Symmetric Difference — प्रत्येक के नीचे पूरा Step-by-Step Solution दिया गया है।
Question: If A = {1, 2, 3} and B = {2, 3, 4}, find A ∪ B, A ∩ B, A – B.
A ∪ B = {1, 2, 3, 4}
A ∩ B = {2, 3}
A – B = {1}
Question: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A Δ B (Symmetric Difference).
A Δ B = (A – B) ∪ (B – A)
A – B = {1, 2}, B – A = {5, 6} → A Δ B = {1, 2, 5, 6}
Question: What is the power set of A = {a, b}?
P(A) = {∅, {a}, {b}, {a, b}} , |P(A)| = 2² = 4
Question: Which of the following is correct?
(1) A ∪ ∅ = ∅ (2) A ∪ ∅ = A (3) A ∩ U = ∅ (4) A ∩ U = U
Question: If A is any set and U is universal set, then A ∩ A' = ?
Question: De Morgan's law states that:
(1) (A ∪ B)' = A' ∪ B'
(2) (A ∪ B)' = A' ∩ B'
(3) (A ∩ B)' = A' ∩ B'
(4) (A ∪ B)' = A ∪ B
Question: A ∩ (B ∪ C) equals:
(1) (A ∩ B) ∪ (A ∩ C)
(2) (A ∪ B) ∩ (A ∪ C)
(3) (A ∩ B) ∩ (A ∩ C)
(4) (A ∪ B) ∪ (A ∪ C)
Question: Identify which one of the following statements is INCORRECT regarding Set Theory?
(1) Intersection of two disjoint sets is a null set.
(2) Complement of a Universal set is a null set.
(3) If X ∩ Y = φ, then X - Y ≠ X
(4) X - Y = X ∩ \(\overline{Y}\)
Correct Answer: (3) If X ∩ Y = φ, then X - Y ≠ X
Detailed Explanation:
• (1) Correct: Disjoint sets have no common elements → intersection φ.
• (2) Correct: Complement of Universal set has no elements → φ.
• (3) Incorrect: If X ∩ Y = φ, then X – Y = X (क्योंकि Y के कोई elements X में नहीं हैं) अतः X - Y = X होता है, ≠ X नहीं।
• (4) Correct: X - Y = X ∩ \(\overline{Y}\) standard identity है।
Question: Match the economists with their field of contribution in mathematical economics:
| (A) Slutsky | (I) Input output analysis |
| (B) Dantzig | (II) Linear programming |
| (C) Arrow | (III) CES production function |
| (D) Leontief | (IV) Consumer behaviour analysis |
• Slutsky → Consumer behaviour analysis (Slutsky equation)
• Dantzig → Linear programming (Simplex method)
• Arrow → CES production function
• Leontief → Input output analysis
Question: If A, B and C are three different non-empty sets, then find which of the following are correct in laws of set operation:
i. \(A \cap \phi = \phi\)
ii. \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
iii. \(A \cap A' = \phi\)
iv. \(B - A = B' \cap A'\)
v. \((A \cup B)' = A' \cup B'\)
Correct: i, ii, iii are correct
Incorrect: iv and v
Explanation: i (Intersection with empty set), ii (Distributive Law), iii (Complement law) → सही; iv सही identity \(B - A = B \cap A'\) है; v सही De Morgan \((A \cup B)' = A' \cap B'\) है।
Question: A budget set in consumer theory is an example of:
(1) A finite set (2) A convex set (3) A null set (4) A non-convex set
Question: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find |A ∪ B|.
Question: If A ⊆ B, then which of the following is true?
(1) A ∪ B = A (2) A ∪ B = B (3) A ∩ B = A (4) Both (2) and (3)
Question: If A and B are disjoint sets and A = {1, 2, 3}, B = {4, 5, 6}, then find A ∩ B and A – B.
Question: If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, then A Δ B = ?
📌 नोट : उपरोक्त 15 प्रश्नों में वे सभी UGC NET वाले प्रश्न शामिल हैं जो पिछली बार दिए गए थे (प्रश्न 8, 9, 10) तथा साथ ही Basic से Advanced तक के सभी कॉन्सेप्ट उदाहरण। किसी भी प्रश्न / शब्द को नहीं हटाया गया है।
✅ Quick Revision Table – Set Theory for UGC NET
| Topic | Formula / Rule |
|---|---|
| Union | A ∪ B = {x : x∈A or x∈B} |
| Intersection | A ∩ B = {x : x∈A and x∈B} |
| Complement | A' = U – A |
| Difference | A – B = A ∩ B' |
| Cardinality | |A ∪ B| = |A| + |B| – |A ∩ B| |
| Disjoint | A ∩ B = ∅ |
| De Morgan 1 | (A ∪ B)' = A' ∩ B' |
| De Morgan 2 | (A ∩ B)' = A' ∪ B' |
| Distributive | A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| Power Set Size | |P(A)| = 2^{|A|} |
| Convex Set | λx + (1-λ)y ∈ S for λ∈[0,1] |
📚 अन्य प्रश्न कहाँ मिलेंगे? (Additional Resources)
| Source | What you get |
|---|---|
| IFAS Publications | 2011-2024 PYQs Unit-wise sorted (Mathematical Economics) |
| Prepp.in | 2009-2025 Question Papers PDF (Free Download) |
| Examrace | Official PYPs (1995-2025) Free PDF |
| Atlantic Publishers | 2005-2016 Solved Papers |
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