JEE Mains 2026 Revision Capsule – Mathematics: Sets (Chapter 1)
A sharp, exam-ready revision of Sets for JEE Mains 2026. All definitions, formulas, Venn diagrams, and must-remember shortcuts included in crisp form.
1. What is a Set?
A set is a well-defined collection of distinct objects. Examples: A = {1, 2, 3}, B = {x | x is an even natural number}.
“Well-defined” means everyone interpreting the description will agree on the membership.
2. Types of Sets
• Finite set – limited elements
• Infinite set – unending elements
• Singleton set – {a}
• Empty set – ∅
• Equal sets – same elements
• Equivalent sets – equal number of elements (same cardinality)
Note: JEE often asks whether ∅ ⊂ A or ∅ ∈ A — they are different ideas.
3. Subsets
A ⊆ B → every element of A is in B
A ⊂ B → A is a proper subset of B
Total subsets of a set of n elements: 2ⁿ
Power set: P(A) = set of all subsets of A
4. Operations on Sets
Union (A ∪ B): in A or B
Intersection (A ∩ B): common elements
Difference (A − B): in A but not B
Complement (A′): not in A (relative to universal set U)
Identities: A ∪ ∅ = A, A ∩ ∅ = ∅, A ∪ A′ = U, A ∩ A′ = ∅
5. Venn Diagram Formulas
Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Three sets:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
These appear nearly every year in JEE Mains.
6. De Morgan’s Laws
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
7. JEE High-Yield Shortcuts
• “At least one” → union
• “Exactly one” → symmetric difference A ⊕ B
• A ⊕ B = (A − B) ∪ (B − A)
• Disjoint sets → A ∩ B = ∅
8. Common JEE Question Types
✔ Find n(A ∪ B) using given n(A), n(B), n(A ∩ B)
✔ Count subsets for a given n
✔ Simplify set expressions
✔ Solve 3-set Venn problems
✔ Identify valid/invalid set definitions
9. Quick Practice (Self-check)
1. How many proper subsets does a set with 5 elements have?
2. If n(B)=50 and A ⊂ B, find maximum n(A).
3. Evaluate (A ∩ B′) ∪ (A′ ∩ B).
4. If A and B are disjoint, what is n(A ∪ B)?
5. Find n(A ∪ B ∪ C) given individual counts.
Sets are the grammar of mathematics. Mastering them sharpens logic for everything ahead — relations, functions, algebra, and probability.
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