Introduction
A set is a well-defined collection of distinct objects called elements. Georg Cantor (1845-1918) is the father of set theory.
Representation of Sets
Sets can be represented in two forms:
(a) Roster/Tabular Form: A = {1, 2, 3, 4, 5}.
(b) Set Builder Form: A = {x : x is a natural number, x ≤ 5}.
Symbols: ∈ means 'belongs to', ∉ means 'does not belong to'.
Types of Sets
Empty Set (∅ or {}), Finite Set, Infinite Set, Equal Sets (A = B if same elements), Equivalent Sets (same cardinality), Universal Set (U), Singleton Set.
Subsets and Power Set
A ⊆ B if every element of A is in B. Number of subsets of set with n elements = 2ⁿ.
Power Set P(A) = set of all subsets of A.
Operations on Sets
Union (A ∪ B), Intersection (A ∩ B), Difference (A - B or A\B), Complement (A').
| Operation | Symbol | Meaning | Example (A={1,2,3}, B={2,3,4}) |
|---|---|---|---|
| Union | ∪ | All elements in A or B | {1,2,3,4} |
| Intersection | ∩ | Common elements | {2,3} |
| Difference | A-B | In A but not B | {1} |
| Complement | A' | Not in A (wrt U) | Depends on U |
| Symmetric Diff | A△B | In A or B but not both | {1,4} |
Venn Diagrams
Venn Diagrams are visual representations of sets.
The universal set is usually represented by a rectangle, and its subsets by circles within the rectangle. Union is the entire area of two circles combined. Intersection is the overlapping region. Difference A-B is the region in circle A that doesn't overlap with circle B.
De Morgan's Laws
De Morgan's Laws are fundamental rules linking set operations:
1. (A ∪ B)' = A' ∩ B'
2. (A ∩ B)' = A' ∪ B'
De Morgan's Laws are extremely important for JEE Main. Always verify: (A ∪ B)' ≠ A' ∪ B' — this is a common mistake.
Solved Examples
(a) If A = {1,2,3} and B = {2,3,4,5}, find A∪B and A∩B.
A∪B = {1,2,3,4,5}
A∩B = {2,3}
(b) If n(A)=7, n(B)=9, n(A∩B)=3, find n(A∪B).
n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 7 + 9 - 3 = 13.
(c) Find the power set of A = {a,b}.
P(A) = {∅, {a}, {b}, {a,b}}.