Introduction to Advanced Mathematics 1st Edition by Randall Holmes ISBN ZLIBIO1534892

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ISBN 13: ZLIBIO1534892 
Author: Randall Holmes
A comprehensive textbook designed to bridge the gap between elementary mathematical concepts and more advanced mathematical theory. Aimed at students transitioning from introductory to higher-level mathematics, it covers a broad range of topics such as set theory, logic, functions, proofs, and the foundations of mathematical reasoning. The book emphasizes rigorous thinking, problem-solving techniques, and the development of abstract mathematical concepts, providing readers with the tools needed to approach more specialized fields in mathematics with confidence and clarity.

Introduction to Advanced Mathematics 1st Edition Table of contents:

Part I: Foundations of Mathematics

Chapter 1: Sets and Set Theory

• Basic Definitions and Notations
• Set Operations: Union, Intersection, and Difference
• Venn Diagrams and Set Identities
• Cartesian Products and Relations
• Functions as Relations

Chapter 2: Logic and Proof Techniques

• Propositional Logic
• Logical Connectives and Truth Tables
• Methods of Proof: Direct, Indirect, Contradiction, and Counterexamples
• Quantifiers and Predicate Logic
• Mathematical Induction

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Part II: Functions and Structures

Chapter 3: Functions and Mappings

• Definition and Types of Functions
• One-to-One, Onto, and Inverses
• Composition of Functions
• Function Notation and Operations
• The Concept of Cardinality

Chapter 4: Integers and Mathematical Structures

• The Integers and Divisibility
• Greatest Common Divisors and Least Common Multiples
• Prime Numbers and Fundamental Theorem of Arithmetic
• Modular Arithmetic
• Introduction to Algebraic Structures: Groups, Rings, and Fields

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Part III: Abstract Reasoning and Number Systems

Chapter 5: Rational and Real Numbers

• Properties of Rational Numbers
• Real Number System and Completeness
• Algebraic and Transcendental Numbers
• Decimals and the Decimal Expansion of Real Numbers
• The Axiomatic Approach to the Real Numbers

Chapter 6: Structures and Groups

• Introduction to Group Theory
• Group Axioms and Examples
• Subgroups, Cosets, and Lagrange’s Theorem
• Cyclic Groups and Symmetry Groups
• Group Homomorphisms and Isomorphisms

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Part IV: Advanced Mathematical Concepts

Chapter 7: Introduction to Proofs in Advanced Topics

• The Role of Proofs in Advanced Mathematics
• Proof Strategies for Advanced Topics
• Examples from Number Theory and Analysis
• Common Pitfalls and How to Avoid Them

Chapter 8: Introduction to Real Analysis

• The Concept of Limits
• Continuity and Discontinuity
• The Fundamental Theorem of Calculus
• Sequences and Series of Real Numbers
• Convergence and Divergence

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Part V: Advanced Topics in Mathematical Logic

Chapter 9: Set Theory and Cardinality

• Infinite Sets and Countability
• Cantor’s Theorem and Uncountable Sets
• Zermelo-Fraenkel Set Theory
• The Axiom of Choice and its Implications
• Ordinals and Cardinals in Advanced Set Theory

Chapter 10: Introduction to Combinatorics

• Basic Counting Principles
• Permutations and Combinations
• Binomial Theorem and Pascal’s Triangle
• Pigeonhole Principle
• Introduction to Graph Theory

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