## Proof and the Art of Mathematics

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Contents (pg. vii)
Preface (pg. xiii)
A Note to the Instructor (pg. xvii)
A Note to the Student (pg. xxi)
1. A Classical Beginning (pg. 1)
1.1 The number 2 is irrational (pg. 2)
1.2 Lowest terms (pg. 4)
1.3 A geometric proof (pg. 5)
1.4 Generalizations to other roots (pg. 6)
Mathematical Habits (pg. 7)
Exercises (pg. 8)
2. Multiple Proofs (pg. 9)
2.1 n2-n is even (pg. 10)
2.2 One theorem, seven proofs (pg. 10)
2.3 Different proofs suggest different generalizations (pg. 12)
Mathematical Habits (pg. 13)
Exercises (pg. 14)
Credits (pg. 14)
3. Number Theory (pg. 15)
3.1 Prime numbers (pg. 15)
3.2 The fundamental theorem of arithmetic (pg. 16)
3.3 Euclidean division algorithm (pg. 19)
3.4 Fundamental theorem of arithmetic, uniqueness (pg. 21)
3.5 Infinitely many primes (pg. 21)
Mathematical Habits (pg. 24)
Exercises (pg. 25)
4. Mathematical Induction (pg. 27)
4.1 The least-number principle (pg. 27)
4.2 Common induction (pg. 28)
4.3 Several proofs using induction (pg. 29)
4.4 Proving the induction principle (pg. 32)
4.5 Strong induction (pg. 33)
4.6 Buckets of Fish via nested induction (pg. 34)
4.7 Every number is interesting (pg. 37)
Mathematical Habits (pg. 37)
Exercises (pg. 38)
Credits (pg. 39)
5. Discrete Mathematics (pg. 41)
5.1 More pointed at than pointing (pg. 41)
5.2 Chocolate bar problem (pg. 43)
5.3 Tiling problems (pg. 44)
5.4 Escape! (pg. 47)
5.5 Representing integers as a sum (pg. 49)
5.6 Permutations and combinations (pg. 50)
5.7 The pigeon-hole principle (pg. 52)
5.8 The zigzag theorem (pg. 53)
Mathematical Habits (pg. 55)
Exercises (pg. 55)
Credits (pg. 56)
6. Proofs without Words (pg. 57)
6.1 A geometric sum (pg. 57)
6.2 Binomial square (pg. 58)
6.3 Criticism of the "without words'' aspect (pg. 58)
6.4 Triangular choices (pg. 59)
6.5 Further identities (pg. 60)
6.6 Sum of odd numbers (pg. 60)
6.7 A Fibonacci identity (pg. 61)
6.8 A sum of cubes (pg. 61)
6.9 Another infinite series (pg. 62)
6.10 Area of a circle (pg. 62)
6.11 Tiling with dominoes (pg. 63)
6.12 How to lie with pictures (pg. 66)
Mathematical Habits (pg. 68)
Exercises (pg. 69)
Credits (pg. 70)
7. Theory of Games (pg. 71)
7.1 Twenty-One (pg. 71)
7.2 Buckets of Fish (pg. 73)
7.3 The game of Nim (pg. 74)
7.4 The Gold Coin game (pg. 79)
7.5 Chomp (pg. 81)
7.6 Games of perfect information (pg. 83)
7.7 The fundamental theorem of finite games (pg. 85)
Mathematical Habits (pg. 89)
Exercises (pg. 89)
Credits (pg. 90)
8. Pick's Theorem (pg. 91)
8.1 Figures in the integer lattice (pg. 91)
8.2 Pick's theorem for rectangles (pg. 92)
8.3 Pick's theorem for triangles (pg. 93)
8.4 Amalgamation (pg. 95)
8.5 Triangulations (pg. 97)
8.6 Proof of Pick's theorem, general case (pg. 98)
Mathematical Habits (pg. 98)
Exercises (pg. 99)
Credits (pg. 100)
9. Lattice-Point Polygons (pg. 101)
9.1 Regular polygons in the integer lattice (pg. 101)
9.2 Hexagonal and triangular lattices (pg. 104)
9.3 Generalizing to arbitrary lattices (pg. 106)
Mathematical Habits (pg. 107)
Exercises (pg. 108)
Credits (pg. 110)
10. Polygonal Dissection Congruence Theorem (pg. 111)
10.1 The polygonal dissection congruence theorem (pg. 111)
10.2 Triangles to parallelograms (pg. 112)
10.3 Parallelograms to rectangles (pg. 113)
10.4 Rectangles to squares (pg. 113)
10.5 Combining squares (pg. 114)
10.6 Full proof of the dissection congruence theorem (pg. 115)
10.7 Scissors congruence (pg. 115)
Mathematical Habits (pg. 117)
Exercises (pg. 118)
Credits (pg. 119)
11. Functions and Relations (pg. 121)
11.1 Relations (pg. 121)
11.2 Equivalence relations (pg. 122)
11.3 Equivalence classes and partitions (pg. 125)
11.4 Closures of a relation (pg. 127)
11.5 Functions (pg. 128)
Mathematical Habits (pg. 129)
Exercises (pg. 130)
12. Graph Theory (pg. 133)
12.1 The bridges of Königsberg (pg. 133)
12.2 Circuits and paths in a graph (pg. 134)
12.3 The five-room puzzle (pg. 137)
12.4 The Euler characteristic (pg. 138)
Mathematical Habits (pg. 139)
Exercises (pg. 140)
Credits (pg. 142)
13. Infinity (pg. 143)
13.1 Hilbert's Grand Hotel (pg. 143)
Hilbert's bus (pg. 144)
Hilbert's train (pg. 144)
Hilbert's half marathon (pg. 145)
Cantor's cruise ship (pg. 146)
13.2 Countability (pg. 146)
13.3 Uncountability of the real numbers (pg. 150)
Alternative proof of Cantor's theorem (pg. 152)
Cranks (pg. 153)
13.4 Transcendental numbers (pg. 154)
13.5 Equinumerosity (pg. 156)
13.6 The Shröder-Cantor-Bernstein theorem (pg. 157)
13.7 The real plane and real line are equinumerous (pg. 159)
Mathematical Habits (pg. 160)
Exercises (pg. 160)
Credits (pg. 161)
14. Order Theory (pg. 163)
14.1 Partial orders (pg. 163)
14.2 Minimal versus least elements (pg. 164)
14.3 Linear orders (pg. 166)
14.4 Isomorphisms of orders (pg. 167)
14.5 The rational line is universal (pg. 168)
14.6 The eventual domination order (pg. 170)
Mathematical Habits (pg. 171)
Exercises (pg. 171)
15. Real Analysis (pg. 173)
15.1 Definition of continuity (pg. 173)
15.2 Sums and products of continuous functions (pg. 175)
15.3 Continuous at exactly one point (pg. 177)
15.4 The least-upper-bound principle (pg. 178)
15.5 The intermediate-value theorem (pg. 178)
15.6 The Heine-Borel theorem (pg. 179)
15.7 The Bolzano-Weierstrass theorem (pg. 181)
15.8 The principle of continuous induction (pg. 182)
Mathematical Habits (pg. 185)
Exercises (pg. 185)
Credits (pg. 187)
Answers to Selected Exercises (pg. 189)
Bibliography (pg. 199)
Index of Mathematical Habits (pg. 201)
Notation Index (pg. 203)
Subject Index (pg. 205)
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