(EBOOK PDF)A Concise Introduction to Pure Mathematics Fourth Edition by Martin Liebeck 9781498722957 1498722954 full chapters

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• ISBN 13:9781498722957

• Author:Martin Liebeck 

 A Concise Introduction to Pure Mathematics
Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Fourth Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Euler’s formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions. New to the Fourth Edition Two new chapters that serve as an introduction to abstract algebra via the theory of groups, covering abstract reasoning as well as many examples and applications New material on inequalities, counting methods, the inclusion-exclusion principle, and Euler’s phi function Numerous new exercises, with solutions to the odd-numbered ones Through careful explanations and examples, this popular textbook illustrates the power and beauty of basic mathematical concepts in number theory, discrete mathematics, analysis, and abstract algebra. Written in a rigorous yet accessible style, it continues to provide a robust bridge between high school and higher-level mathematics, enabling students to study more advanced courses in abstract algebra and analysis.

A Concise Introduction to Pure Mathematics Fourth Table of contents:

 Preface
New Features of the Fourth Edition
Chapter 1 Sets and Proofs
Sets
Proofs
Example 1.1
Example 1.2
Example 1.3
Example 1.4
Example 1.5
Example 1.6
Quantifiers
Exercises for Chapter 1
Chapter 2 Number Systems
The Real Numbers
RULES 2.1
Rationals and Irrationals
PROPOSITION 2.1
PROPOSITION 2.2
PROPOSITION 2.3
PROPOSITION 2.4
Example 2.1
PROPOSITION 2.5
Exercises for Chapter 2
Chapter 3 Decimals
PROPOSITION 3.1
PROPOSITION 3.2
Example 3.1
PROPOSITION 3.3
PROPOSITION 3.4
Example 3.2
PROPOSITION 3.5
Eercises for Chapter 3
Chapter 4 nth Roots and Rational Powers
PROPOSITION 4.1
PROPOSITION 4.2
Exercises for Chapter 4
Chapter 5 Inequalities
RULES 5.1
Example 5.1
Example 5.2
Example 5.3
Example 5.4
Example 5.5
Example 5.6
Example 5.7
Example 5.8
Example 5.9
Example 5.10
Example 5.11
Example 5.12
Example 5.13
Example 5.14
Example 5.15
Exercises for Chapter 5
Chapter 6 Complex Numbers
Geometrical Representation of Complex Numbers
Example 6.1
De Moivre’s Theorem
THEOREM 6.1 (De Moivre’s Theorem)
PROPOSITION 6.1
Example 6.2
Example 6.3
Example 6.4
Example 6.5
The eiθ Notation
PROPOSITION 6.2
Roots of Unity
PROPOSITION 6.3
Example 6.6
Example 6.7
Exercises for Chapter 6
Chapter 7 Polynomial Equations
Solution of Cubic Equations
Example 7.1
Higher Degrees
The Fundamental Theorem of Algebra
THEOREM 7.1 Fundamental Theorem of Algebra
THEOREM 7.2
THEOREM 7.3
Relationships between Roots
PROPOSITION 7.1
Example 7.2
Exercises for Chapter 7
Chapter 8 Induction
Principle of Mathematical Induction
Example 8.1
Example 8.2
Principle of Mathematical Induction II
Example 8.3
Example 8.4
Guessing the Answer
Example 8.5
The Σ Notation
Example 8.6
Example 8.7
Geometric Examples
Example 8.8
Example 8.9
Prime Factorization
PROPOSITION 8.1
Principle of Strong Mathematical Induction
Example 8.10
Cauchy’s Inequality
PROPOSITION 8.2
Example 8.11
Example 8.12
Exercises for Chapter 8
Chapter 9 Euler’s Formula and Platonic Solids
THEOREM 9.1
THEOREM 9.2
Regular and Platonic Solids
THEOREM 9.3
Exercises for Chapter 9
Chapter 10 The Integers
PROPOSITION 10.1
PROPOSITION 10.2
The Euclidean Algorithm
Example 10.1
THEOREM 10.1
PROPOSITION 10.3
Example 10.2
PROPOSITION 10.4
PROPOSITION 10.5
PROPOSITION 10.6
Exercises for Chapter 10
Chapter 11 Prime Factorization
The Fundamental Theorem of Arithmetic
THEOREM 11.1 (Fundamental Theorem of Arithmetic)
Some Consequences of the Fundamental Theorem
PROPOSITION 11.1
PROPOSITION 11.2
PROPOSITION 11.3
PROPOSITION 11.4
Example 11.1
Exercises for Chapter 11
Chapter 12 More on Prime Numbers
THEOREM 12.1
THEOREM 12.2
Exercises for Chapter 12
Chapter 13 Congruence of Integers
PROPOSITION 13.1
Example 13.1
PROPOSITION 13.2
Arithmetic with Congruences
PROPOSITION 13.3
PROPOSITION 13.4
Example 13.2
Example 13.3
Example 13.4
Example 13.5
Example 13.6
PROPOSITION 13.5
Congruence Equations
Example 13.7
PROPOSITION 13.6
The System ℤm
Example 13.8
Example 13.9
Exercises for Chapter 13
Chapter 14 More on Congruence
Fermat’s Little Theorem
THEOREM 14.1 (Fermat’s Little Theorem)
PROPOSITION 14.1
Finding kth Roots Modulo m
Example 14.1
PROPOSITION 14.2
PROPOSITION 14.3
Finding Large Primes
PROPOSITION 14.4
Example 14.2
Exercises for Chapter 14
Chapter 15 Secret Codes
RSA Codes: Encoding
Decoding
Security
A Little History
Exercises for Chapter 15
Chapter 16 Counting and Choosing
Example 16.1
THEOREM 16.1 (Multiplication Principle)
Example 16.2
PROPOSITION 16.1
Binomial Coefficients
PROPOSITION 16.2
Example 16.3
Example 16.4
THEOREM 16.2 Binomial Theorem
PROPOSITION 16.3
Ordered Selections
Example 16.5
PROPOSITION 16.4
Example 16.6
Multinomial Coefficients
Example 16.7
PROPOSITION 16.5
Example 16.8
THEOREM 16.3 Multinomial Theorem
Example 16.9
Exercises for Chapter 16
Chapter 17 More on Sets
Unions and Intersections
Example 17.1
PROPOSITION 17.1
Example 17.2
Cartesian Products
The Inclusion—Exclusion Principle
PROPOSITION 17.2
Example 17.3
THEOREM 17.1 Inclusion Exclusion Principle
Example 17.4
PROPOSITION 17.3
PROPOSITION 17.4
Exercises for Chapter 17
Chapter 18 Equivalence Relations
Example 18.1
Equivalence Classes
Example 18.2
Example 18.3
PROPOSITION 18.1
Exercises for Chapter 18
Chapter 19 Functions
DEFINITION
Example 19.1
DEFINITION
PROPOSITION 19.1
The Pigeonhole Principle
Example 19.2
Inverse Functions
DEFINITION
Example 19.3
Composition of Functions
DEFINITION
Example 19.4
PROPOSITION 19.2
Counting Functions
PROPOSITION 19.3
Exercises for Chapter 19
Chapter 20 Permutations
Example 20.1
PROPOSITION 20.1
Example 20.2
Composition of Permutations
Example 20.3
Four Fundamental Features
PROPOSITION 20.2
The Cycle Notation
PROPOSITION 20.3
Example 20.4
Example 20.5
Repeating a Permutation
PROPOSITION 20.4
Example 20.6
Example 20.7
Even and Odd Permutations
Example 20.8
DEFINITION
PROPOSITION 20.5
PROPOSITION 20.6
PROPOSITION 20.7
Example 20.9
Example 20.10
Exercises for Chapter 20
Chapter 21 Infinity
DEFINITION
PROPOSITION 21.1
Example 21.1
Countable Sets
DEFINITION
Example 21.2
PROPOSITION 21.2
PROPOSITION 21.3
PROPOSITION 21.4
Example 21.3
An Uncountable Set
THEOREM 21.1
A Hierarchy of Infinities
DEFINITION
DEFINITION
PROPOSITION 21.5
Exercises for Chapter 21
Chapter 22 Introduction to Analysis: Bounds
Uper and Lower Bounds
DEFINITION
Example 22.1
DEFINITION
Example 22.2
COMPLETENESS AXIOM
Example 22.3
Exercises for Chapter 22
Chapter 23 More Analysis: Limits
Example 23.1
DEFINITION
Example 23.2
Example 23.3
Example 23.4
Bounded Sequences
PROPOSITION 23.1
Calculating Limits
PROPOSITION 23.2
Example 23.5
Increasing and Decreasing Sequences
DEFINITION
PROPOSITION 23.3
Example 23.6
Exercises for Chapter 23
Chapter 24 Yet More Analysis: Continuity
Continuous Functions
DEFINITION
Example 24.1
PROPOSITION 24.1
Example 24.2
The Intermediate Value Theorem
THEOREM 24.1 Intermediate Value Theorem
Existence of nth Roots
PROPOSITION 24.2
A Special Case of the Fundamental Theorem of Algebra
PROPOSITION 24.3
Exercises for Chapter 24
Chapter 25 Introduction to Abstract Algebra: Groups
Definition and Examples of Groups
Example 25.1
DEFINITION
DEFINITION
Example 25.2
DEFINITION
First Results
PROPOSITION 25.1
PROPOSITION 25.2
Multiplicative Notation for Groups
PROPOSITION 25.3
Example 25.3
Exercises for Chapter 25
Chapter 26 Introduction to Abstract Algebra: More on Groups
Subgroups
DEFINITION
Example 26.1
PROPOSITION 26.1
Example 26.2
PROPOSITION 26.2
DEFINITION
Example 26.3
DEFINITION
Order of an Element
DEFINITION
Example 26.4
PROPOSITION 26.3
Example 26.5
PROPOSITION 26.4
Lagrange’s Theorem
THEOREM 26.1 (Lagrange’s Theorem)
Consequences of Lagrange’s Theorem
COROLLARY 26.1
COROLLARY 26.2
COROLLARY 26.3
Example 26.6
Applications to Number Theory
PROPOSITION 26.5
Example 26.7
THEOREM 26.2
PROPOSITION 26.6
Example 26.8
PROPOSITION 26.7
PROPOSITION 26.8
DEFINITION
PROPOSITION 26.9
DEFINITION
PROPOSITION 26.10
Example 26.9
Proof of Lagrange’s Theorem
DEFINITION
Example 26.10
PROPOSITION 26.11
PROPOSITION 26.12
PROPOSITION 26.13
Completion of the proof
Exercises for Chapter 26
Back Matter
Solutions to Odd-Numbered Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Further Reading
Index of Symbols

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