Class 11 Sets Revision Notes PDF

## Class 11 - Chapter 1 - Sets - Revision Class ### NCERT Topics 1. Introduction 2. Sets and their Representations 3. The Empty Set 4. Finite and Infinite Sets 5. Equal Sets 6. Subsets 7. Power Set 8. Universal Set 9. Venn Diagrams 10. Operations on Sets 11. Complement of a Set 12. Practical Probl...

## Class 11 - Chapter 1 - Sets - Revision Class ### NCERT Topics 1. Introduction 2. Sets and their Representations 3. The Empty Set 4. Finite and Infinite Sets 5. Equal Sets 6. Subsets 7. Power Set 8. Universal Set 9. Venn Diagrams 10. Operations on Sets 11. Complement of a Set 12. Practical Problems on Union and Intersection of Two Sets ### Topics * Roster and Set Builder * Types of Sets * Subsets & Superset * Proper and Improper Subsets * Power Set * Intervals * Venn Diagram * Set Operations * De Morgan's Law ### What is a Set? * Set is a well-defined collection of objects. * Elements in a set are not repeated. * A set can be represented in two forms: Roster and Set-Builder. * Set is represented by a capital letter, and the elements are written between curly brackets. * Example: A = {1, 2, 3, 4} ### Representation of Sets * **N:** Set of all Natural Numbers * N = {1, 2, 3, 4, ...} * **W:** Set of all Whole Numbers * W = {0, 1, 2, 3, 4, ...} * **Z:** Set of all Integers * Z = {... -3, -2, -1, 0, 1, 2, 3 ...} * **Q:** Set of all Rational Numbers * Q = {..., -1/3, 1/7, -15, 0.25, 0.99, ..., 7/4, 9/5, ...} * **R:** Set of all Real Numbers * R = {..., π, √2, 1.732, -4, -5π, √2, ...} ### Roster & Set-Builder * **Roster:** * All elements are in {}. * Example: Set of all Natural Numbers below 6. * A = {1, 2, 3, 4, 5} * **Set-Builder:** * Elements are based on some rules and characteristics. * Example: * x is a natural number such that. * A = {x: x ∈ N, x < 6} ### Power Set * Power set is the set of all subsets (including empty set Φ). * The number of elements in a power set with n elements is 2^n. * Example: * A = {a, b, c} (n(A) = 3). * Power set of A = P(A). * P(A) = {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, Φ} * n(P(A)) = 2^3 = 8 ### Proper & Improper Subsets * **Proper Subset:** Will not be equal to the superset. * **Improper Subset:** Will be equal to the superset. ### Universal Set * In a particular context, we have to deal with elements and subsets of a basic set which is relevant to that context. * The universal set is usually denoted by U. * It is the superset of all sets. ### Venn Diagrams * U: Universal set (U = {1, 2, 3, 4, 6, 7, 8, 9, 10}). * A: Set A (A = {1, 2, 3, 4}). * B: Set B (B = {3, 6, 7, 8}). * C: Set C (C = {10}). This is a singleton set, which means there's only one element in the set. ### Set Operations * **Union:** AUB - Addition/Combination. * **Intersection:** A∩B - Common. * **Difference:** AB - Subtraction. * **Complement:** A' - Opposite. ### Union * AUB = {x: x ∈ A or x ∈ B}. * n(AUB) = n(A) + n(B) - n(A∩B). * Properties: * AUB = BUA (commutative). * (AUB)UC = AU(BUC) (associative). * AUΦ = A. * AUA = A. * UUA = U. ### Intersection * A∩B = {x: x ∈ A and x ∈ B}. * **And, Both, Neither.** * Properties: * A∩B = B∩A. * (A∩B)∩C = A∩(B∩C). * Φ∩A = Φ, U∩A = A. * A∩A = A. * A∩(BUC) = (A∩B)U(A∩C). * A∩B = Φ, so A and B are disjoint, and n(A∩B) = 0. ### Difference of Sets * A-B = {x: x ∈ A and x ∉B}. * The difference implies the set of elements which belong to A but not to B. * Properties: * A-B ≠ B-A. * The sets A-B, B-A and A∩B are disjoint. * A-B ⊆ A, and B-A ⊆ B. * A - Φ = A, and A - A = Φ. * A - B = A - (A∩B). ### Complement * A' = {x: x ∈ U and x ∉ A}. * A' = U-A. * n(A') = n(U) - n(A). * **Properties:** * AUA' = U. * A∩A' = Φ. * (A')' = A. * Φ' = U. * U' = Φ. ### De Morgan's Law * (AUB)' = A'∩B' * (A∩B)' = A'UB' ### Practical Problems * n(AUB) = n(A) + n(B) - n(A∩B) * If A and B are disjoint: n(AUB) = n(A) + n(B) * n(A-B) = n(A) - n(A∩B) * n(A') = n(U) - n(A) * n(AUBUC) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C)