Index
a
- Abelian group 386
- Abel, Niels 391
- absolute value, complex number 285
- abstract algebra 369–370
- abstraction 370
- additive identity 434
- adjacency matrix 329–330
- aleph null 151
- algebra 370
- algebraically closed field 285
- algebraic field 270, 437–438
- algebraic functions 228
- algebraic group 377
- algebraic numbers 163–165
- algebra of relations 192
- algebra of sets 123
- antecedent/premise sentence 25
- antisymmetric relation 197–198
- Aquinas, Thomas 143
- Argand, Jean Robert 284
- Aristotle 3, 143
- arithmetic axioms for real numbers 270–271
- arithmetic in modular arithmetic 222
- arithmetization of analysis 40, 75
- Ars Magna 281
- Artin, Emil 274
- associative operation 386, 433
- assumption sentence 25
- atomic sentences 6
- axiomatic set theory 105
- axiom of choice (AC)
- axioms 52, 136
b
- backwards compositions 235
- backwards proof 57–59
- Bell number 139, 215
- Bernoulli, John 90
- biconditional sentence 30–32
- Bieberbach, L. 396
- big number 129
- bijection function 149
- binary operation
- binary relation 184
- binomial theorem 48
- Birkhoff, Garrett 202
- blood typing 194
- Bolzano, Bernard 277
- Boolean algebras 11
- Boolean field 272, 279
- Boole, George 11, 105, 202
- boundary point 356
- bounds on an open interval 201–202
- Burke, Edmund 195
- Burnside’s lemma 56
c
- Cantor–Bernstein theorem 170–171
- Cantor, George 105, 146, 156, 167, 350
- Cantorian set theory 172
- Cantor’s diagonalization theorem 157–158
- Cantor’s discovery of larger sets 167–170
- Cantor set 364–365
- Cantor’s power set theorem 168–169
- Cantor’s seminal contribution to infinity 146
- Cantor’s seminal theorem 61
- Cardano, Gerolamo 281
- cardinality of sets
- cardinality of the continuum 159
- cardinal number 153
- Carmichael totient function conjecture 240
- Cartesian product 181–183, 191, 418
- Catalan numbers 140
- Cauchy cycle notation 408
- Cauchy, Louis 357, 409
- Cayley table 378, 398
- chromatic number of a graph 318–319
- circle 105
- classroom function 235
- closed operations 270
- closed set 363
- codomain of the function 225
- Cohen, Paul 172
- combinations
- common rings 435–437
- commutative algebraic system 377, 386
- commutative group 433
- commutative rings 433
- compact sets 124
- complement of a set 99–103
- completeness axioms 270, 274
- complete ordered field
- algebraic field 278
- arithmetic axioms for real numbers 270–271
- Boolean field 272, 279
- completeness axioms 270, 274
- complex numbers 272
- conventions and notation 271–272
- field axioms 269
- least upper bound and greatest lower bounds 274–277
- not an ordered field 279
- order axioms 269
- ordered fields 273–274, 279
- rational functions 272
- rational numbers 272
- well‐ordering principle 279
- well‐ordering theorem 279
- complex addition 289–290
- complex division 291–292
- complex exponential 286, 288
- complex multiplication 291
- complex numbers 272
- as an algebraic field 283–284
- basic arithmetic of 289–292
- to Cartesian form 296
- complex exponential and Euler’s theorem 286–288
- complex variables in polar form 288–289
- definition 282–283
- de Moivre’s formula 297
- fractional powers 297
- imaginary numbers and two dimensions 284–285
- introduction 281–282
- polar coordinates 285–286
- to polar form 296
- primitive roots of unity 297
- roots and powers of a 292–295
- complex subtraction 290
- compositions
- compound sentences 6–11
- conclusion sentence 25
- conditional sentence 24–27
- congruence classes 217
- congruent modulo 216
- conjecture 58
- conjugate of complex number 285
- conjunctions 43–45
- conjunctive normal forms (CNF) 16–17
- connected sets 253
- consequent sentence 25
- consistent axioms 105
- continuity
- continuous images of intervals 251
- continuum hypothesis (CH) 171–172
- contradiction 12, 14–15, 18
- coprime 240
- corollary 56
- cosets 427–429, 432
- countably infinite sets 151–154
- counting
- Bell number 139
- Catalan numbers 140
- combinations 131–135
- distinguishable permutations 130–131, 138
- famous apple problem 140
- functions 138
- lottery problem 141
- multiplication principle 126–127
- permutations 127–128
- permutations of racers 128–130
- pigeonhole principle 135–138
- pizza cutter’s formula 141
- relatively prime hard 142
- relatively prime light 142
- relatively prime medium 142
- round robin tournament 141
- single elimination tournament 139
- Snail Darter Society 139
- world series time 139
- counting functions 127
- counting partitions 215
- counting subsets 126–127
- cycle notation 416
- cyclic group 391–393
d
- Decartes, Rene 27, 282
- Dedekind cut
- Dedekind, Richard 202, 263, 433
- de Fermat, Pierre 90
- degrees of symmetry 372
- de la Vallee Poussin 60
- delta 357
- δ‐neighborhood 350
- de Moivre’s formula 297
- DeMorgan, Augustus 11, 83
- DeMorgan’s laws 12–13, 353
- denial of sentences 19
- dense orders 209–210
- descent proof 83
- difference of two sets 104
- digital logical circuits I, 22
- digital logical circuits II, 23
- digraph. see directed graphs
- dihedral group 396–397
- dihedral group D2 382
- dihedral multiplication table 382
- Diophantine equation 77–78
- directed edges 321
- directed graphs 186–188
- direct predecessor 321
- direct product of groups 399, 418
- direct proofs 54–56, 65, 88
- direct successor 321
- Dirichlet definition of a function 224–225
- Dirichlet, Peter Gustav Lejeune 138, 224–225, 227
- Dirichlet principle. see pigeonhole principle
- Dirichlet’s function 252–253
- disjoint sets 104, 112
- disjoint subsets 273
- disjunctions 43–45
- disjunctive normal forms (DNF) 16–17, 20–21
- distinguishable permutations 130–131, 138
- distributive laws 33
- distributive operation 434
- division property 197–198
- domain of a composition 229
- domain of the function 225
- domain of the relation 188
- dominance graphs
- double‐holed torus 347
- doughnut and coffee cup 335, 336, 345
- drawer principle 138
- dynamic programming 327–329, 332
e
- Einstein, Albert 4
- empty set 99–103
- enantiomorphic shape 378
- epsilon 357
- equality of sets 99–103
- equilateral triangle
- equivalence classes 214–215
- in the plane 218–219
- equivalence relation
- in analysis 220
- arithmetic in modular arithmetic 222
- in calculus 220
- counting 221
- counting partitions 215
- definition 212–213
- equivalence classes in logic 221
- equivalence sets of polynomials 221
- finding equivalence classes 220
- finding the 220
- modular arithmetic 216–219, 221
- partitioning property of the 214–215
- partition of a set 213–214
- similar matrices 221
- unusual 220
- equivalence sets of polynomials 221
- equivalence, tautology, and contradiction 12
- equivalent intervals 159–160
- equivalent sets 147, 154
- Euclidean geometry 334
- Euclid’s proof 61
- Euler characteristic
- Euler cycle 303
- Euler diagram 25–26
- Euler, Leonard 60, 99, 227, 301
- Euler paths 313–315
- and circuits 303–304
- Euler’s characteristic for planar graphs 309–311
- Euler’s conjecture 80
- Euler’s formula 346
- Euler’s original graph theorem 304
- Euler’s proof of the PNT 60
- Euler’s theorem 48, 287
- Euler totient function 68, 240
- Euler tour 304, 311–313
- even and odd natural numbers 154
- even integer 54
- exclusive OR, 19
- existential quantifiers 38–39
- experimental sciences 54
- extended laws for sets 119–120
- exterior point of set 356
f
- factor group 429
- families of sets
- algebra of sets 123
- compact sets 124
- extended laws for sets 119–120
- identity of an indexed family 123
- indexed family 118
- index set 115
- infinite intersections and unions 116–117
- infinite intersections of unions 117–118
- in the plane 122–123
- set projection 118–119
- sets of length zero 123–124
- topologies on a set 120–121, 124
- unions and intersections of 116, 122
- famous identity, mathematical induction 85–86
- Fermat’s last theorem 48
- Ferrers diagram 210–211
- Fibonacci sequence 93
- field
- field axioms 269
- fields of functions 272
- finite cyclic group 391, 392
- finite field 438–440
- finite group 386
- first‐order logic. see predicate logic
- first‐stage dominances 323
- Fischer, M.E., 125
- fluent 62
- fluxion 62
- Fourier, Joseph 227
- four symmetries, rectangle 374, 375
- Fraenkel, Abraham 105, 172, 173
- Frege, Gottlob 43
- Frege–Russell thesis 78
- functional analysis 232
- functional equation 237
- function relation
- backwards compositions 235
- brief history of the 227
- Carmichael totient function conjecture 240
- classroom function 235
- composition of operators 236–237
- compositions 228–232, 235
- counting functions 239, 240
- Dirichlet definition 224–225
- Euler totient function 240
- examples 226
- functional equation 237
- graphing 234
- graphing a composition 236
- injections, surjections, bijections 237, 238
- inverse functions 232–234, 238
- mystery function 234
- as ordered pairs 238
- recursive function 237
- relation definition of a 227–228
- shifting domain of a composition 236
- testing 234
- functions of functions 232
- fundamental theorem of arithmetic 59, 89, 90–91
g
- Galileo 3, 143, 227
- Galois, Evariste 385, 391
- Galois fields 438
- Galois finite fields 272
- game time, directed graphs 333
- Gauss, Carl 282
- Gauss, Carl Friedrich 217
- Gauss, Karl Friedrich 60
- generalized integers 433
- generator 391
- geometric principle by induction 92
- geometric topology
- geometry 391
- Gödel, Kurt 172
- Gödel’s incompleteness theorem 172
- Goldbach conjecture 58
- Google’s PageRank system 325
- Google’s search engine models 325, 327
- graph 182
- graph of the function 225
- graph theory
- chromatic number of a graph 318–319
- definition 302
- Euler circuits 303–304
- Euler paths 303–304, 313–315
- Euler’s characteristic for planar graphs 309–311
- Euler tour 311–313
- knight’s tour 317–318
- Konigsberg bridge problem 304–306
- main ingredients of 302–303
- Moser spindle 319
- Platonic solids 316
- regular graph 318
- weighted graphs 307–308
- greatest lower bound (glb) 200–204
- grid points 258–259
- group dominance 330, 331
- group leader 324
- group theory
- guaranteed subset 101
h
- Hadamard, Jacques 60, 232
- Hamiltonian graph 313–315
- Hamiltonian tour 313–315
- Hasse diagrams and directed graphs 199–200, 205, 206, 207
- Hausdorff, Felix 199
- head/source 321
- Heisenberg group 399
- higher‐order relations 188
- Hilbert, David 45, 170, 396
- homeomorphic sets 336
- homeomorphism
- homomorphisms 401–402
- Huygens, Christian 305
i
- identity 386, 387
- symmetry 376
- If and Only If theorems 80–81
- image of a set
- image of the function 225
- imaginary numbers and two dimensions 284–285
- imaginary part of the complex number 284
- incompleteness theorem 45
- in‐degree 322
- independent axioms 105
- indexed family 118
- index set 115
- indirect proofs 54
- induction in calculus 86–87
- inequality by induction 87–88
- inference 27
- infinite arbitrary sets 151
- infinite group 401
- infinite intersections and unions 116–117
- infinite intersections of unions 117–118
- infinite number of prime numbers 59–60
- infinite order 386
- infinite sets 154
- injection 230–231
- injective function 148, 154
- injective relation 192
- integers 258–260
- integrated circuit graph 309–311
- interior point of set 355
- intermediate value theorem 48
- International Congress of Mathematicians 396
- Internet Research 333, 348
- interpretation of images 251
- intersection of a family 116
- intersection of sets 104
- intersections and unions, image of a set 245–250
- inverse 386, 387
- of permutations 408
- inverse functions 232–234, 238
- inverse image of a set 243
- inverse images in topology 249–250
- inverse relations 189, 192–193
- irrational number 165, 166, 265
- isomorphic groups 393–395, 394, 400, 418
- isomorphism 394, 395
k
l
- Lagrange, Joseph‐Louis 391, 422
- Lagrange’s Theorem 422
- Landau’s theorem 332
- Latin square 388, 401
- lattice 207–209
- law of the excluded middle 13, 74
- law of the syllogism 28–29
- least upper bound (lub) 200–204
- Legendre, A.M., 60
- Leibniz, Gottfried Wilhelm 10, 227, 305
- lemma 56
- Lévy, Paul 232
- Liar paradox 6
- limit point 361–362
- Lincoln, Abraham 380
- line symmetry 371
- Liouville constant 163, 166
- Liouville, Joseph 166, 385
- Listing, Johann 342
- logical AND, 7
- for IP addresses 20
- logical connectives 6
- logical disjunction 8
- logically equivalent sentences 12
- logical OR, 7
- logical sentences from truth tables 15
- logicism 43
- logistic thesis 78
- lower bounds 200–204
- Lukasiewicz, Jan 36
m
- Markov Chain 327
- Mary, Clever 91–92
- mathematical induction
- mathematical proofs
- analysis of 54–55
- axioms 52
- counterexample 66
- direct proof 65
- divisibility 65–66
- Euler’s totient function 68
- modus operandi for proving theorems 55–62
- necessary and sufficient conditions 62–65
- Pick’s amazing formula 68–69
- proposition 52
- syllogisms 67–68
- theorem 52
- twin prime conjecture 69–70
- types of 53–54
- mathematics, definitions 3–4
- Maurolico, Francesco 90
- maximal element 201
- Mersenne primes 98
- minimal element 201
- minimum spanning tree 307
- mirror symmetry 371
- Mirzakhani, Maryam 4
- Möbius band 344–345
- Mobius strip 335, 342, 344–345
- modern algebra 370
- modular algebra 441, 442
- modular arithmetic 216–219, 221
- modulo 3 field 441
- modulo 4 multiplication 400
- modulo 5 multiplication 399
- modus operandi, for proving theorems 55–62
- Modus Ponens 31, 36
- Modus Tollens 36
- Moser spindle 319
- multiplication principle 126–127
- multiplication rule 126
- multiplicative identity 434, 440
- multiplicative inverse 441
- multiplying groups 397
- multiplying permutations 406–407
n
- naive set theory 105
- vs. axiomatic set theory 105
- n‐ary relation 188
- NASC for disjoint sets 112
- necessary and sufficient conditions (NASC) 62–65
- negation of quantified propositions 42–43
- negative integers 260
- Noether, Emmy 377, 433
- non‐Cantorian set theory 172
- nonconvex polyhedra 346
- nonequivalence relations 213, 219–220
- nonnegative integers 260
- nonnegative real number 285
- nonobvious statement 35–36
- nonzero members 273
- normal subgroups 428
- NOT operator 7
- null set. see empty set
- number theory 391
o
- octic group 424–426
- odd integer 54
- one‐to‐one correspondence function 149, 230
- one‐to‐one function 148, 230
- onto function 148–149, 230
- open interval, bounds on an 201–202
- open set 102, 120
- order 386
- order axioms 269
- ordered fields 273–274, 279
- order matters 41–42
- order relations
- complex numbers 206–207
- composition of partial orders 210
- definition 195–196
- dense orders 209–210
- division property 197–198
- finding relations 204
- functions 204
- Hasse diagrams and directed graphs 199–200, 205, 206, 207
- inverse of a partial order 210
- lattice of partitions 209
- lattices 208–209
- partially ordered sets 198
- partitions of a natural number 210–211
- sups and infs 205
- symmetric relation 199
- testing 204
- total order and symmetric relations 198–199
- upper and lower bounds 205
- upper bounds, lower bounds, glb, and lub 200–204
- ordinal number 153
- out‐degree 322
p
- p‐adic number fields 272
- PageRank system 325–327, 326
- partially ordered sets 198, 202
- partial order 195–196
- partition function 210
- partition of a set 213–214
- Pascal, Blaise 90
- Peano, Giuseppe 78, 90, 104, 228
- Peano’s axioms 90, 93
- Peirce, Benjamin 25
- Peirce, Charles Saunders 202
- Perelman, Grigori 58
- permutations 127–128
- alternating group 415
- composition of 416–417
- cycle notation for 408–409
- decomposition of 417
- distinguishable 130–131, 138
- finding 416
- identity 416
- inverses of 408
- mapping 403, 404
- matrices 418
- product of 405–407
- of racers 128–130
- symmetric group S3 414–415
- symmetric group Sn 413
- transposition 411–413
- visualization of 404
- Pick’s amazing formula 68–69
- Pierce, Charles Saunders 56, 78
- pigeonhole principle 135–138
- pizza cutter’s formula 141
- planar graphs, Euler’s characteristic for 309–311
- plane symmetry 374
- platonic solids 316, 341
- Poincare Conjecture 58
- points 351
- point‐set topology
- polar coordinates 285–286
- polyhedron 341
- Post, Emil 25
- power set 101
- predicate logic
- predicates 38
- prime number 54
- prime number theorem (PNT) 59
- principal root 294
- principle argument of complex number 286
- principle of mathematical induction 83, 84
- product of permutations
- proof by contradiction 60–62
- proof by contrapositive 56–57
- proof by demonstration 73
- proof by induction 88
- proofs in predicate logic
- proofs, mathematical
- analysis of 54–55
- axioms 52
- counterexample 66
- direct proof 65
- divisibility 65–66
- Euler’s totient function 68
- modus operandi for proving theorems 55–62
- necessary and sufficient conditions 62–65
- Pick’s amazing formula 68–69
- proposition 52
- syllogisms 67–68
- theorem 52
- twin prime conjecture 69–70
- types of 53–54
- proper subgroups 420
- proper subset 99
- propositions 38. See also sentences
- negation of quantified 42–43
- Pythagoras’proof 61
- Pythagorean theorem 78
q
r
- radial symmetry 372
- range of the function 225
- range of the relation 188
- ranking webpages 333
- rational functions 272
- rationalizing the denominator 292
- rational numbers 153–154, 272
- and the completeness axiom 277
- rationals 260–262
- real analysis 43, 102
- real numbers
- arithmetic axioms for 270–271
- arithmetic of the 266
- building of the 258
- construction of the integers 258–260
- construction of the rationals 260–262
- decimal representations 267
- decimal to fractions 267
- Dedekind cut 263–266
- definition 262–263
- equivalence relation 266–267
- irrational number 267–268
- mathematical carpentry 257
- synthetic approach 257
- real part of the complex number 284
- recursive function 237
- reductio ad absurdum 55
- reflections symmetry 374
- reflective symmetry 371, 373
- reflexive relation 197, 212
- regular graph 318
- regular polygons 396
- relations
- algebra of 192
- binary 183–184, 193–194
- blood typing 194
- Cartesian product identities 193
- composition of 189–190, 193
- counting 193
- definition 184
- directed graphs 186–188
- domain and range of a 188
- graphing a 191
- graphing relations and their inverses 193
- identifying 185–186
- important types of 192
- inverse 192–193
- inverses and compositions 189
- meaning of 192
- naming a 192
- number of 193
- typical 185
- relatively prime group 393, 395, 400
- residue classes 217
- ring
- Rota, GianCarlo 51
- rotational symmetry
- rotation permutation 411–412
- round‐robin tournaments 322, 331
- “rubber‐sheet” geometry 334
s
- same cardinality/cardinality number 147
- satisfiable sentence 50
- Schubfachprinzip 138
- second‐order logic 56, 78
- second‐stage dominances 324
- sentences
- sentential logic
- set inclusion 104
- set intersection 104
- set projection 118–119
- sets
- computer representation of 113–114
- De Morgan’s laws 108–109, 114
- difference between 112
- distributive law 112
- vs. logic 107–108
- logic, and arithmetic 109
- members, and subsets 111
- and membership 97–99
- NASC for disjoint sets 112
- notation 109–110
- power sets 111
- relations with truth tables 113
- union, intersection, and difference of 103–105
- universe, subset, equality, complement, empty 99–103
- venn diagrams of 105–106
- set theory, need for axioms in 172–173
- set union 104
- simple sentences 6, 17
- six roots of unity 294
- Snail Darter Society 139
- social networking, dominance graphs 322–324
- spanning tree 307
- special ring 435, 436, 442
- strict order 196, 273
- strong induction 88, 89–91
- subgroup
- Cayley table 430, 431
- center of a group 431
- cosets 427–429, 432
- of cyclic groups 426–427
- definition 420
- generated by R240 431
- Hasse diagram for 431
- Klein four‐group 422
- matrix 431
- proper 420
- quotient group 429–430
- rotational symmetries of equilateral triangle 420
- of symmetries 420–421
- test of 422–426
- of Z6 430
- of Z8 431
- of Z11 431
- subset 99–103
- surjection 232
- surjective function 148–149, 154
- surjective relation 192
- syllogisms 22
- symbols 4
- symmetric group S2 417
- symmetric group S3 414–415
- symmetric group Sn 403, 413
- symmetric relation 198–199, 212
- symmetry
- axes 380
- in calculus 371
- definition of 374
- of differential equation 384
- of ellipse 382
- of equilateral triangle 377–381
- groups 383
- mapping/transformation 373–374
- observations 377
- of parallelogram 382
- of rectangle 374–377
- rotational (see rotational symmetry)
- of square 383
- of tetrahedron 383
- in two dimensions 371–373
t
- tail/sink 321
- tautology 12, 13–15, 18
- ternary relation 188
- theorem 52
- theory of algebraic equations 391
- third‐stage dominances 325
- three‐valued logic 36
- tiny topologies 121
- topological chemistry 338
- topological fingerprints 334, 335
- topological invariants 339–340
- topologically equivalent, definition 335
- topologically equivalent objects
- topological properties 335
- topological space 358–360, 359
- topologies on a set 120–121, 124
- topology 338, 342
- toroidal polyhedra 346–347
- totality of elements 99
- total order 273
- and symmetric relations 198–199
- tournament graphs. see dominance graphs
- transcendental functions 228
- transcendental numbers 163–165
- transfinite numbers 169
- transition matrix 326
- transitive relation 198, 212
- transitive subsets 101–102
- translations symmetry 374
- transposition 411–413, 417
- transpositions commute 417
- tree 307
- triangle graph 309–310
- triple‐hole torus 347, 348
- truncated cube 346
- truncated solid 346
- truth tables 18
- twin prime conjecture 69–70
- two‐cycle 411–413, 417
- type 1 Dedekind cut 264
- type 2 Dedekind cut 264
- type 3 Dedekind cut 264–265
- typical relation 185
u
- uncountable cardinality 156
- uncountable set 156, 159
- algebraic numbers 165
- all lines are equal 160
- Cantor’s diagonalization theorem 157–158
- Cantor’s surprise 161–165
- cardinality of functions 165
- countable plus singleton 165
- definition 156
- equivalent intervals 159–160
- irrational numbers 165, 166
- Liouville constant 166
- more real numbers than natural numbers 158–159
- unending interesting properties of numbers 75–77
- union of a family 116
- union of sets 103–104
- unions and intersections 116, 122
- unity identity 434
- universal direct proof 72–73
- universal quantifier 38–39, 71
- universe set 99–103
- unusual equivalence relation 220
- upper bounds 200–204
- Urysohn’s lemma 56
v
- variable quantity 227
- variables 38
- Veblen, Oswald 273
- Venn diagram 25
- Venn, George 105
- visual proof 86
- Volterra, Vito 232
- Vorstudien zur Topologie 342
w
x
- Xenocrates 405
z
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