Contents

Preface
1     Let’s Play!
1.1     A Direct Approach
1.2     Fibonacci Numbers and the Golden Ratio
1.3     Inductive Reasoning
1.4     Natural Numbers and Divisibility
1.5     The Primes
1.6     The Integers
1.7     The Rationals, the Reals, and the Square Root of 2
1.8     End-of-Chapter Exercises
2     Discovering and Presenting Mathematics
2.1     Truth, Tabulated
2.2     Valid Arguments and Direct Proofs
2.3     Proofs by Contradiction
2.4     Converse and Contrapositive
2.5     Quantifiers
2.6     Induction
2.7     Ubiquitous Terminology
2.8     The Process of Doing Mathematics
2.9     Writing Up Your Mathematics
2.10   End-of-Chapter Exercises
3     Sets
3.1     Set Builder Notation
3.2     Sizes and Subsets
3.3     Union, Intersection, Difference, and Complement
3.4     Many Laws and a Few Proofs
3.5     Indexing
3.6     Cartesian Product
3.7     Power
3.8     Counting Subsets
3.9     A Curious Set
3.10   End-of-Chapter Exercises
4     The Integers and the Fundamental Theorem of Arithmetic
4.1     The Well-Ordering Principle and Criminals
4.2     Integer Combinations and Relatively Prime Integers
4.3     The Fundamental Theorem of Arithmetic
4.4     LCM and GCD
4.5     Numbers and Closure
4.6     End-of-Chapter Exercises
5     Functions
5.1     What is a Function?
5.2     Domain, Codomain, and Range
5.3     Injective, Surjective, and Bijective
5.4     Composition
5.5     What is a Function? Redux!
5.6     Inverse Functions
5.7     Functions and Subsets
5.8     A Few Facts About Functions and Subsets
5.9     End-of-Chapter Exercises
6     Relations
6.1     Introduction to Relations
6.2     Partial Orders
6.3     Equivalence Relations
6.4     Modulo m
6.5     Modular Arithmetic
6.6     Invertible Elements
6.7     End-of-Chapter Exercises
7     Cardinality
7.1     The Hilbert Hotel, Count von Count, and Cookie Monster
7.2     Cardinality
7.3     Countability
7.4     Key Countability Lemmas
7.5     Not Every Set is Countable
7.6     Using the Schröder–Bernstein Theorem
7.7     End-of-Chapter Exercises
8     The Real Numbers
8.1     Completeness
8.2     The Archimedean Property
8.3     Sequences of Real Numbers
8.4     Geometric Series
8.5     The Monotone Convergence Theorem
8.6     Famous Irrationals
8.7     End-of-Chapter Exercises
9     Probability and Randomness
9.1     A Class of Lyin’ Weasels
9.2     Probability
9.3     Revisiting Combinations
9.4     Events and Random Variables
9.5     Expected Value
9.6     Flipped or Faked?
9.7     End-of-Chapter Exercises
10   Algebra and Symmetry
10.1   An Example from Modular Arithmetic
10.2   The Symmetries of a Square
10.3   Group Theory
10.4   Cayley Tables
10.5   Group Properties
10.6   Isomorphism
10.7   Isomorphism and Group Properties
10.8   Examples of Isomorphic and Non-isomorphic Groups
10.9   End-of-Chapter Exercises
11   Projects
11.1   The Pythagorean Theorem
11.2   Chomp and the Divisor Game
11.3   Arithmetic–Geometric Mean Inequality
11.4   Complex Numbers and the Gaussian Integers
11.5   Pigeons!
11.6   Mirsky’s Theorem
11.7   Euler’s Totient Function
11.8   Proving the Schröder–Bernstein Theorem
11.9   Cauchy Sequences and the Real Numbers
11.10 The Cantor Set
11.11 Five Groups of Order 8
Solutions, Answers, or Hints to In-Text Exercises
Bibliography
Index