Sets

Board: CBSE | Class: Class 11

Comprehensive study notes for Sets by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Definition and Representation

A set is a well-defined collection of objects. Represented as: Roster form {a, b, c} or Set-builder form {x: property of x}. Null set/Empty set: set with no elements, denoted by φ or {}.

Types of Sets

Finite set: Countable number of elements. Infinite set: Uncountable. Equal sets: Same elements. Equivalent sets: Same cardinality. Subset: A ⊂ B (all elements of A are in B). Power set: Set of all subsets, cardinality = 2ⁿ.

Operations on Sets

Union: A ∪ B = elements in A or B or both. Intersection: A ∩ B = elements in both. Difference: A – B = elements in A but not in B. Complement: A’ or U – A.

Venn Diagrams

Universal set U is represented by a rectangle. Sets are circles. Overlapping regions show intersections. Disjoint sets have no overlap.

De Morgan's Laws

Law 1: (A ∪ B)’ = A’ ∩ B’. Law 2: (A ∩ B)’ = A’ ∪ B’. These relate complements of unions/intersections.

Cardinality (n(A))

n(A ∪ B) = n(A) + n(B) – n(A ∩ B). n(A ∪ B ∪ C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C).

Important Formulas

Subset	A ⊂ B if x ∈ A ⇒ x ∈ B
Union	A ∪ B = {x: x ∈ A or x ∈ B}
Intersection	A ∩ B = {x: x ∈ A and x ∈ B}
Complement	A' = {x ∈ U: x ¬in; A}
De Morgan 1	(A ∪ B)' = A' ∩ B'
De Morgan 2	(A ∩ B)' = A' ∪ B'
Cardinality	n(A∪B)=n(A)+n(B)-n(A∩B)

Solved Examples

Example 1: If A = {1,2,3,4}, B = {3,4,5,6}, find A∪B, A∩B, A-B.

Solution: A∪B = {1,2,3,4,5,6}. A∩B = {3,4}. A-B = {1,2}.

Example 2: If U = {1,…,10}, A = {2,4,6,8,10}, find A’.

Solution: A’ = {1,3,5,7,9}.

Example 3: Verify De Morgan: U={1..10}, A={2,4,6}, B={3,6,9}.

Solution: (A∪B)’ = {1,5,7,8,10}. A’∩B’ = {1,5,7,8,10}. Verified.

Practice Questions

1. If A = {x: x is a prime ≤ 10}, B = {x: x is even ≤ 10}. Find A∪B, A∩B.
2. In a class of 60, 35 play cricket, 30 play football, 20 play both. How many play neither?
3. Prove De Morgan’s first law using Venn diagrams.
4. Find the power set of {a, b, c}.
5. If n(A)=25, n(B)=20, n(A∩B)=8, find n(A∪B).

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