دانلود کتاب Introduction to Proofs and Proof Strategies (Cambridge Mathematical Textbooks)

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Cover
Half-title page
Series page
Title page
Copyright page
Reviews
Contents
List of Symbols and Notation
Preface
Part I Core Material
1 Numbers, Quadratics and Inequalities
1.1 The Quadratic Formula
1.2 Working with Inequalities – Setting the Stage
1.3 The Arithmetic-Geometric Mean and the Triangle Inequalities
1.4 Types of Numbers
1.5 Problems
1.6 Solutions to Exercises
2 Sets, Functions and the Field Axioms
2.1 Sets
Notation and Terminology
The Interval Notation and Set Operations
2.2 Functions
2.3 The Field Axioms
The Real Numbers System
2.4 Appendix: Infinite Unions and Intersections
2.5 Appendix: Defining Functions
2.6 Problems
2.7 Solutions to Exercises
3 Informal Logic and Proof Strategies
3.1 Mathematical Statements and their Building Blocks
Quantifiers
Connectives
3.2 The Logic Symbols
3.3 Truth and Falsity of Compound Statements
Implications
A Remark on Quantifiers
3.4 Truth Tables and Logical Equivalences
Equivalences Involving Quantifiers
3.5 Negation
3.6 Proof Strategies
Direct Proof
Proof by Contrapositive
Proof by Contradiction
3.7 Problems
3.8 Solutions to Exercises
4 Mathematical Induction
4.1 The Principle of Mathematical Induction
4.2 Summation and Product Notation
4.3 Variations
Variation 1
Variation 2
Variation 3
4.4 Additional Examples
Recursive Definitions of Sequences
A Fallacy
4.5 Strong Mathematical Induction
The Principle of Strong Mathematical Induction
Proof of Existence
Proof of Uniqueness
4.6 Problems
4.7 Solutions to Exercises
5 Bijections and Cardinality
5.1 Injections, Surjections and Bijections
5.2 Compositions
5.3 Cardinality
5.4 Cardinality Theorems
5.5 More Cardinality and the Schröder–Bernstein Theorem
5.6 Problems
5.7 Solutions to Exercises
6 Integers and Divisibility
6.1 Divisibility and the Division Algorithm
Proof of Uniqueness
6.2 Greatest Common Divisors and the Euclidean Algorithm
The Euclidean Algorithm
6.3 The Fundamental Theorem of Arithmetic
6.4 Problems
6.5 Solutions to Exercises
7 Relations
7.1 The Definition of a Relation
Symbols for Commonly Used Relations
7.2 Equivalence Relations
7.3 Equivalence Classes
7.4 Congruence Modulo n
7.5 Problems
7.6 Solutions to Exercises
Part II Additional Topics
8 Elementary Combinatorics
8.1 Counting Arguments: Selections, Arrangements and Permutations
8.2 The Binomial Theorem and Pascal’s Triangle
8.3 The Pigeonhole Principle
8.4 The Inclusion-Exclusion Principle
8.5 Problems
8.6 Solutions to Exercises
9 Preview of Real Analysis – Limits and Continuity
9.1 The Limit of a Sequence
9.2 The Limit of a Function
One-Sided Limits
9.3 The Relation between Limits of Functions and Sequences
9.4 Continuity and Differentiability
Continuity
Differentiability
9.5 Problems
9.6 Solutions to Exercises
10 Complex Numbers
10.1 Background
10.2 The Field of Complex Numbers
10.3 The Complex Plane and the Triangle Inequality
Sums of Complex Numbers
The Triangle Inequality
10.4 Square Roots and Quadratic Equations
Complex Quadratic Equations
10.5 Polar Representation of Complex Numbers
Geometric Interpretation of Products
10.6 De Moivre’s Theorem and Roots
Arbitrary Roots of Complex Numbers
General Polynomial Equations
10.7 The Exponential Function
10.8 Problems
10.9 Solutions to Exercises
11 Preview of Linear Algebra
11.1 The Spaces R[sup(n)] and their Properties
11.2 Geometric Vectors
Scalar Multiplication
11.3 Abstract Vector Spaces
11.4 Subspaces
11.5 Linear Maps and Isomorphisms
11.6 Problems
11.7 Solutions to Exercises
Index