1 - Sets
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:
No natural number between 1 and 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.