This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
Author(s): Richard Hammack
Edition: 3.3
Publisher: Richard Hammack
Year: 2020
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
Limit 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
