Chapter 1: Sets

1.1 Introduction

• The concept of a set is fundamental in modern mathematics, utilized across various branches such as geometry, sequences, and probability.

• Developed by the mathematician Georg Cantor (1845-1918) while addressing problems on trigonometric series.

• Focus of the chapter: basic definitions and operations involving sets.

1.2 Sets and Their Representations

• Everyday examples of sets include collections of objects (e.g., a pack of cards, a cricket team).

• Mathematical examples:

  • Odd natural numbers less than 10: {1, 3, 5, 7, 9}

  • Rivers in India

  • Vowels in the English alphabet: {a, e, i, o, u}

  • Types of triangles

  • Prime factors of 210: {2, 3, 5, 7}

  • Solutions of the equation x² - 5x + 6 = 0: {2, 3}

• A well-defined collection allows determining membership (e.g., the river Ganga belongs to rivers in India, but the Nile does not).

Key Points:

• Synonyms: Objects, elements, members of a set.

• Denotation: Sets are represented by capital letters (A, B, C), and elements by lowercase letters (a, b, c).

• Membership notation:

  • a ∈ A (a belongs to A)

  • b ∉ A (b does not belong to A)

• Set Representation Methods:

  1. Roster/Tabular Form: List elements within braces:

     • E.g., Set of even positive integers less than 7: {2, 4, 6}.

  2. Set-Builder Form: Describes the property common to all elements:

     • E.g., V = {x : x is a vowel in the English alphabet}.

1.3 The Empty Set

• Definition: A set containing no elements is called the empty set (denoted by φ or {}).

• Examples of empty sets:

  • A = {x : 1 < x < 2 and x is a natural number} (no such element exists).

• Properties:

  • B = {x : x is a student in both Classes X and XI} is an empty set.

1.4 Finite and Infinite Sets

• Finite Set: Contains a definite number of elements.

  • E.g., A = {1, 2, 3, 4, 5} has 5 elements.

• Infinite Set: Contains an unlimited number of elements.

  • E.g., set of natural numbers.

• Examples:

  • W (days of the week) is finite.

  • Set of points on a line is infinite.

1.5 Equal Sets

• Definition: Two sets A and B are equal if they contain exactly the same elements (A = B).

• Example: A = {1, 2, 3, 4} and B = {4, 3, 2, 1} are equal sets.

1.6 Subsets

• Definition: A set A is a subset of B if every element of A is also in B (A ⊆ B).

• Example: If A = {2, 3} and B = {2, 3, 4}, then A ⊆ B.

• Properties:

  • Empty set is a subset of every set: φ ⊆ A.

  • A set is a subset of itself: A ⊆ A.

1.7 Universal Set

• Definition: The set containing all objects within a particular context.

• Denoted by U.

• Example: In number theory, U could be all integers.

1.8 Venn Diagrams

• Used to represent relationships between sets.

• Universal set represented as a rectangle; subsets as circles.

1.9 Operations on Sets

• Union: A ∪ B = {elements in A or B (including both)}.

• Intersection: A ∩ B = {elements common to both A and B}.

• Difference: A - B = {elements in A not in B}.

• Example: If A = {1, 2, 3} and B = {2, 3, 4}, then:

  • A ∪ B = {1, 2, 3, 4}

  • A ∩ B = {2, 3}

  • A - B = {1}

1.10 Complement of a Set

• Definition: Complement A' contains elements not in A with respect to universe U.

• Example: If U = {1, 2, 3, 4, 5} and A = {1, 3}, then A' = {2, 4, 5}.

• De Morgan’s Laws:

  • (A ∪ B)' = A' ∩ B'

  • (A ∩ B)' = A' ∪ B'

Summary

• This chapter covers essential definitions and operations of sets, including the concepts of empty sets, finite/infinite sets, equal sets, subsets, and operations like union, intersection, and difference.

Historical Note

• Georg Cantor's pioneering work in set theory laid the foundation for modern mathematics. Key developments in set theory involved recognizing various paradoxes and formalizing axioms to refine the concepts introduced by him.