Chapter 1: Sets

1.1 Introduction

• The concept of set is fundamental in present-day mathematics, used across various branches like geometry, sequences, and probability.

• Developed by the German mathematician Georg Cantor (1845-1918) during his work on trigonometric series.

• Discusses basic definitions and operations involving sets.

1.2 Sets and their Representations

• Everyday collections manifest as sets in mathematics (e.g., odd natural numbers, prime factors).

• Examples of well-defined collections of objects in sets:

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

  • Rivers of India: {Ganga, Yamuna, etc.}

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

  • Triangles: various classifications.

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

• Each example is a well-defined collection; we can determine membership distinctly.

1.3 Well-defined Collections

• An essential aspect is that a set must be well-defined (e.g., the River Nile does not belong to the rivers of India).

• Definitions related to sets:

  • Natural numbers: N

  • Integers: Z

  • Rational numbers: Q

  • Real numbers: R

  • Positive integers: Z+, positive rational numbers: Q+, positive real numbers: R+.

• A collection may not be well-defined if criteria for membership are subjective (e.g., "most renowned mathematicians").

1.4 Terminology

• Terms used in sets:

  • Objects/Elements/Members: synonyms.

  • Denoted by capital letters (A, B)

  • Elements denoted by small letters (a, b).

• Membership notation:

  • If a is in A: a ∈ A.

  • Not in A: b ∉ A.

  • Examples illustrate the membership concept.

1.5 Set Representation Methods

Roster Form

• Elements listed between braces { } and separated by commas.

  • Example: Set of natural numbers dividing 42: {1, 2, 3, 6, 7, 14, 21, 42}.

• Order does not matter, and duplicates are not listed.

Set-builder Form

• Describes a set in terms of a property.

  • Example: V = {x : x is a vowel in the English alphabet}.

• General structure: A = {x : property P of x}.

1.6 Examples of Set Representation

• Roster vs. set-builder forms shown for various sets (natural numbers, vowels, etc.).

• Exercises involving representing sets in both forms.

1.7 The Empty Set

• Definition: A set with no elements is empty, denoted by φ or {}.

• Examples demonstrate empty sets in different contexts:

  1. No natural number between 1 and 2.

  2. Equation not satisfied by rational numbers.

1.8 Finite and Infinite Sets

• Finite set: has a specific number of elements.

  • Example: Set of days in a week.

• Infinite set: does not have a finite number of elements (e.g., natural numbers).

• Representation and counting methods for finite sets and understanding of infinite sets.

1.9 Equal Sets

• Definition of equal sets (A = B).

• Examples to clarify equalities and comparison of sets.

  • Note on repetition and ordering of elements in sets.

  • Identifying pairs of equal sets through membership.

1.10 Subsets

• Definition: A set A is a subset of B (A ⊂ B) if all elements of A are in B.

• General properties of subsets:

  • A is a subset of itself.

  • The empty set is a subset of every set.

  • Proper subsets if A ⊂ B and A ≠ B.

1.11 Intervals as Subsets of R

• Definition of open and closed intervals in R.

• Notation and examples highlighting interval representations.

1.12 Universal Set

• Definition and implications of a universal set.

• Relation to subsets in specific contexts (e.g., all integers as a universal set for primes).

1.13 Venn Diagrams

• Visualization of set relationships using Venn diagrams.

• Understanding of how Venn diagrams represent unions, intersections, and differences of sets.

1.14 Operations on Sets

• Union (∪) and intersection (∩) definitions and properties.

• Difference of sets (A - B) and examples to demonstrate.

1.15 Complement of a Set

• Definition of a complement with respect to the universal set.

• Example showing the calculation of complements and related properties.

1.16 Properties of Set Operations

• Summarizing properties for union, intersection, and complements (including complementary laws and De Morgan’s laws).

1.17 Historical Note

• Overview of contributions by Georg Cantor and subsequent notable mathematicians in developing set theory.

• Evolution and challenges in the formalization of set theory, mentioning Russell's Paradox and axiomatic developments.

Summary

• Consolidation of key points regarding sets, operations, and properties discussed in the chapter.