دانلود کتاب Book of Proof

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Contents
Preface
Introduction
I Fundamentals
1. Sets
1.1 Introduction to Sets
1.2 The Cartesian Product
1.3 Subsets
1.4 Power Sets
1.5 Union, Intersection, Difference
1.6 Complement
1.7 Venn Diagrams
1.8 Indexed Sets
1.9 Sets That Are Number Systems
1.10 Russell\'s Paradox
2. Logic
2.1 Statements
2.2 And, Or, Not
2.3 Conditional Statements
2.4 Biconditional Statements
2.5 Truth Tables for Statements
2.6 Logical Equivalence
2.7 Quantifiers
2.8 More on Conditional Statements
2.9 Translating English to Symbolic Logic
2.10 Negating Statements
2.11 Logical Inference
2.12 An Important Note
3. Counting
3.1 Lists
3.2 The Multiplication Principle
3.3 The Addition and Subtraction Principles
3.4 Factorials and Permutations
3.5 Counting Subsets
3.6 Pascal\'s Triangle and the Binomial Theorem
3.7 The Inclusion-Exclusion Principle
3.8 Counting Multisets
3.9 The Division and Pigeonhole Principles
3.10 Combinatorial Proof
II How to Prove Conditional Statements
4. Direct Proof
4.1 Theorems
4.2 Definitions
4.3 Direct Proof
4.4 Using Cases
4.5 Treating Similar Cases
5. Contrapositive Proof
5.1 Contrapositive Proof
5.2 Congruence of Integers
5.3 Mathematical Writing
6. Proof by Contradiction
6.1 Proving Statements with Contradiction
6.2 Proving Conditional Statements by Contradiction
6.3 Combining Techniques
6.4 Some Words of Advice
III More on Proof
7. Proving Non-Conditional Statements
7.1 If-and-Only-If Proof
7.2 Equivalent Statements
7.3 Existence Proofs; Existence and Uniqueness Proofs
7.4 Constructive Versus Non-Constructive Proofs
8. Proofs Involving Sets
8.1 How to Prove a∈A
8.2 How to Prove A⊆B
8.3 How to Prove A=B
8.4 Examples: Perfect Numbers
9. Disproof
9.1 Disproving Universal Statements: Counterexamples
9.2 Disproving Existence Statements
9.3 Disproof by Contradiction
10. Mathematical Induction
10.1 Proof by Induction
10.2 Proof by Strong Induction
10.3 Proof by Smallest Counterexample
10.4 The Fundamental Theorem of Arithmetic
10.5 Fibonacci Numbers
IV Relations, Functions and Cardinality
11. Relations
11.1 Relations
11.2 Properties of Relations
11.3 Equivalence Relations
11.4 Equivalence Classes and Partitions
11.5 The Integers Modulo n
11.6 Relations Between Sets
12. Functions
12.1 Functions
12.2 Injective and Surjective Functions
12.3 The Pigeonhole Principle Revisited
12.4 Composition
12.5 Inverse Functions
12.6 Image and Preimage
13. Proofs in Calculus
13.1 The Triangle Inequality
13.2 Definition of a Limit
13.3 Limits That Do Not Exist
13.4 Limit Laws
13.5 Continuity and Derivatives
13.6 Limits at Infinity
13.7 Sequences
13.8 Series
14. Cardinality of Sets
14.1 Sets with Equal Cardinalities
14.2 Countable and Uncountable Sets
14.3 Comparing Cardinalities
14.4 The Cantor-Bernstein-Schröder Theorem
Conclusion
Solutions
Index