Book of Proof

Preface
Introduction
I Fundamentals
	Sets
		Introduction to Sets
		The Cartesian Product
		Subsets
		Power Sets
		Union, Intersection, Difference
		Complement
		Venn Diagrams
		Indexed Sets
		Sets That Are Number Systems
		Russell's Paradox
	Logic
		Statements
		And, Or, Not
		Conditional Statements
		Biconditional Statements
		Truth Tables for Statements
		Logical Equivalence
		Quantifiers
		More on Conditional Statements
		Translating English to Symbolic Logic
		Negating Statements
		Logical Inference
		An Important Note
	Counting
		Lists
		The Multiplication Principle
		The Addition and Subtraction Principles
		Factorials and Permutations
		Counting Subsets
		Pascal's Triangle and the Binomial Theorem
		The Inclusion-Exclusion Principle
		Counting Multisets
		The Division and Pigeonhole Principles
		Combinatorial Proof
II How to Prove Conditional Statements
	Direct Proof
		Theorems
		Definitions
		Direct Proof
		Using Cases
		Treating Similar Cases
	Contrapositive Proof
		Contrapositive Proof
		Congruence of Integers
		Mathematical Writing
	Proof by Contradiction
		Proving Statements with Contradiction
		Proving Conditional Statements by Contradiction
		Combining Techniques
		Some Words of Advice
III More on Proof
	Proving Non-Conditional Statements
		If-and-Only-If Proof
		Equivalent Statements
		Existence Proofs; Existence and Uniqueness Proofs
		Constructive Versus Non-Constructive Proofs
	Proofs Involving Sets
		How to Prove aA
		How to Prove AB
		How to Prove A= B
		Examples: Perfect Numbers
	Disproof
		Counterexamples
		Disproving Existence Statements
		Disproof by Contradiction
	Mathematical Induction
		Proof by Induction
		Proof by Strong Induction
		Proof by Smallest Counterexample
		The Fundamental Theorem of Arithmetic
		Fibonacci Numbers
IV Relations, Functions and Cardinality
	Relations
		Relations
		Properties of Relations
		Equivalence Relations
		Equivalence Classes and Partitions
		The Integers Modulo n
		Relations Between Sets
	Functions
		Functions
		Injective and Surjective Functions
		The Pigeonhole Principle Revisited
		Composition
		Inverse Functions
		Image and Preimage
	Proofs in Calculus
		The Triangle Inequality
		Definition of a Limit
		Limits That Do Not Exist
		Limits Laws
		Continuity and Derivatives
		Limits at Infinity
		Sequences
		Series
	Cardinality of Sets
		Sets with Equal Cardinalities
		Countable and Uncountable Sets
		Comparing Cardinalities
		The Cantor-Bernstein-Schröder Theorem
	Conclusion
	Solutions
	Index