HSSLIVE Plus One Maths Chapter 1: Sets Notes

This chapter introduces students to the fundamental concept of sets, which serves as the foundation for much of modern mathematics. Students will learn set notation, operations like union and intersection, and properties of sets including subsets, power sets, and the universal set. Through practical examples and visual representations, this chapter helps students develop logical thinking and classification skills that will be essential for more advanced mathematical concepts.

Chapter 1: Sets

Introduction to Sets

• A set is a well-defined collection of distinct objects
• Objects in a set are called its elements or members
• Notation: Set A = {1, 2, 3, 4, 5}

Representation of Sets

1. Roster Method: Listing all elements within braces
   • Example: A = {1, 2, 3, 4, 5}
2. Set Builder Form: Describing the property of elements
   • Example: A = {x : x is a natural number less than 6}

Types of Sets

• Finite Set: Contains a definite number of elements
  • Example: A = {1, 2, 3, 4, 5}
• Infinite Set: Contains unlimited elements
  • Example: N = {1, 2, 3, …} (set of all natural numbers)
• Empty Set (∅): Contains no elements
  • Example: Set of natural numbers less than 1
• Universal Set (U): Contains all elements under consideration
• Subset (⊆): Every element of set A is also in set B
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B
• Proper Subset (⊂): A is a subset of B, but A ≠ B
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B
• Equal Sets: Two sets with exactly the same elements
  • Example: {1, 2, 3} = {3, 1, 2} (order doesn’t matter)

Set Operations

• Union (A ∪ B): Elements belonging to either A or B or both
  • Example: {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
• Intersection (A ∩ B): Elements common to both A and B
  • Example: {1, 2, 3} ∩ {3, 4, 5} = {3}
• Difference (A – B): Elements in A but not in B
  • Example: {1, 2, 3, 4} – {3, 4, 5} = {1, 2}
• Complement (A’): Elements in the universal set but not in A
  • Example: If U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, then A’ = {2, 4}

Venn Diagrams

• Visual representation of sets and their relationships
• Useful for visualizing set operations

Important Formulas

• 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(A ∩ C) + n(A ∩ B ∩ C)
• n(U) = n(A) + n(A’)