Chapter 1: Sets - CBSE, ICSE, STATE BOARD, JEE Mathematics 📚 Chapter 1: Sets Comprehensive Guide for CBSE, ICSE, STATE BOARD, JEE Mathematic 📖 What is a Set? Definition: A set is a well-defined and distinct collection of objects, known as elements or members. These objects can be anything—numbers, letters, or even other sets. “Well‑defined” means: no confusion about who is in the set. Imagine you have a box · If I say “put all the red apples in the box,” that’s well‑defined. Everyone knows what a red apple is. You can always say “yes, this belongs” or “no, it doesn’t.” · If I say “put the tasty apples in the box,” that’s not well‑defined. What’s tasty to me might not be tasty to you. There’s no clear rule. distinct means no duplicates. Element / Member: Each object of a set is called an element or member of a set. Example : Collection of Natural Numbers. Collection of Whole Numbers. Collection of Vowels. 💡Methods of Representing a Set (i) Roster or Tabular Form: A set is described by listing elements, separated by commas, within braces {}. Example: A = {a, e, i, o, u} Example: B = {1, 2, 3, 4, 5} (ii) Set-builder Form: A set is described by a characterizing property P(x) of its elements x. Notation: {x: P(x) holds} Example: A = {x : x is a natural number and 3 < x < 10} Example: B = {x: x ∈ ℝ, x² - 1 = 0} 🏷️ Types of Sets ✓Empty Set (Null Set) A set having no element in it is called an empty set or void set or null set. φ or ∅ = { } ✓Singleton Set A set containing one element is called a singleton set. Example: A = {5} ✓Finite Set A set having a fixed number of elements is called a finite set. Example: A = {1, 2, 3, 4} has 4 elements ✓Infinite Set A set that is not finite is called an infinite set. Example: ℕ = {1, 2, 3, 4, ...} (Natural Numbers) ✓Equal Sets Two sets A and B are said to be equal if every element of A is a member of B and vice-versa. Set A = { 1,2,3,4 } and B = { 3,4,1,2 } are equal set. A = B ⟺ A ⊆ B and B ⊆ A ⊂ Subsets Definition: A set A is said to be a subset of a set B if every element of A is also an element of B. A ⊆ B ⟺ (a ∈ A ⟹ a ∈ B) 📌Important Notes on Subsets Every set is a subset of itself: A ⊆ A Empty set φ is a subset of every set: φ ⊆ A Number of subsets of a set with n elements = 2ⁿ Example: If A = {1, 2}, then Subset of A is = φ, {1}, {2}, {1,2} 📍Intervals as Subsets of ℝ Let a, b ∈ ℝ and a < b, then: Closed Interval: [a, b] = {x ∈ ℝ : a ≤ x ≤ b} Open Interval: (a, b) = {x ∈ ℝ : a < x < b} Semi-open/Semi-closed: (a, b] = {x ∈ ℝ : a < x ≤ b} Semi-open/Semi-closed: [a, b) = {x ∈ ℝ : a ≤ x < b} ⚡ Power Set & Universal Set 🎯Power Set Definition: The collection of all subsets of set A is called the power set of A, denoted by P(A). Every element in P(A) is a set If A has n elements, then P(A) has 2ⁿ elements Example: If A = {1, 2}, then P(A) = {φ, {1}, {2}, {1,2}} 🌐Universal Set Definition: It is a set which includes all the elements of the sets under consideration. Denoted by U. Example: If A = {1, 2, 3}, B = {3, 4, 7}, C = {2, 8, 9}Then U = {1, 2, 3, 4, 7, 8, 9} ⚙️ Operations on Sets ➕Union of Sets (A ∪ B) Definition: The union of two sets A and B is the set of all those elements which are either in A or in B. Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B. A ∪ B = { 2, 4, 6, 8, 10, 12} In symbols, we write. A ∪ B = { x : x ∈A or x ∈B } Properties of Union: A ∪ B = B ∪ A (Commutative Law) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative Law) A ∪ φ = A (Identity Element) A ∪ A = A (Idempotent Law) U ∪ A = U (Law of U) ∩Intersection of Sets (A ∩ B) Definition: The intersection of two sets A and B is the set of all elements which are common to both A and B. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = { 2, 3, 5, 7 }. Find A ∩ B. A ∩ B = { 2, 3, 5, 7 }. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}. Properties of Intersection: A ∩ B = B ∩ A (Commutative Law) (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative Law) φ ∩ A = φ, U ∩ A = A A ∩ A = A (Idempotent Law) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive Law) ➖Difference of Sets (A - B) Definition: The difference of two sets A and B (A - B) is the set of all those elements of A which do not belong to B. Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. A – B = {1,3,5} and B – A = {8}. A - B = {x: x ∈ A and x ∉ B} Note: A - B ≠ B - A (Not commutative) 🔢 Important Results on Number of Elements 📊What is a Cardinal Number? The cardinal number of a set is simply how many elements are in the set. It’s also called the cardinality.When A = {2, 4, 8, 16, 32, 64}, the cardinality of set A is n(A) = |A| = 6 Example 1:When A = {a, e, i, o, u }, the cardinality of set A is n(A) = |A| = 5 📊Formula for n(A ∪ B) Case 1: If A ∩ B = φ, then n(A ∪ B) = n(A) + n(B) Case 2: If A ∪ B ≠ φ, then n(A ∪ B) = n(A) + n(B) - n(A ∩ B) 📊Formula for n(A ∪ B ∪ C) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C) 🔄 Complement of a Set Definition: Let U be the universal set and let A be a set such that A ⊆ U. The complement of A with respect to U is denoted by A' or A^c and is defined as the set of all those elements of U which are not in A. Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. Find A′, B′. Solution A′ = U - A = {1, 4, 5, 6}, B′ = U - B = { 1, 2, 6 }. A' = {x ∈ U : x ∉ A} ⚖️Properties of Complement Sets Complement Laws: A ∪ A' = U A ∩ A' = φ De Morgan's Laws: (A ∪ B)' = A' ∩ B' (A ∩ B)' = A' ∪ B' Double Complementation: (A')' = A Laws of φ and U: φ' = U and U' = φ 📐 Venn Diagrams Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams use circles (or other shapes) to represent sets and visually show relationships like union, intersection, and complement. 💡 Tip: Venn diagrams are extremely useful for solving complex problems involving multiple sets and for visualizing set operations. ⭐ Quick Reference Symbols ∈ : Element of ∉ : Not an element of ⊆ : Subset of ⊂ : Proper subset of ∪ : Union ∩ : Intersection φ : Empty set U : Universal set 📚 Master the concepts of Sets for JEE Success
