Why Read This Book? - Index of

346 Index
Bijection, 127
Binary operations. See specific binary
operations
Binomial theorem, 157–161
Bolzano-Weierstrass theorem, 200, 201
Boundary, 175–176
Bounded away from L, 171
Bounded functions, 207–208
Bounded sequences, 187–190
Bounded sets, 166
Bounds. See also Least upper bound axiom
from above, 166, 207
from below, 166, 207
greatest lower, 168–169
C
C1-C3 properties (closure of sets), 178.
See also Closure property
Calculus, 165, 190
Cancellation, 49, 51
of addition, 46–47
in domain, 315
multiplicative, 49, 51, 298, 315, 337
in rings, 298, 337
Cardinality, 135, 137, 138
of Cartesian products, 144–148
of finite sets, 138
of infinite sets, 140
Cartesian plane, 112
Cartesian products, 111–112, 144–151
Cauchy sequences, 202–206, 203f, 216
Cayley tables, 245, 246f, 248f, 249f, 253f,
262f, 263f, 274f, 344f
Characteristic zero, 301
Choice, axiom of, 118
Circles, 109
extended real numbers and,
221–222, 221f
unit, 215, 216
Circular addition, 260
Cities example, 92–93, 97–98, 99–101
Clock arithmetic, 260
Closed and open set, 174
Closed intervals, 167
Closed sets, 172–175
Closure property
addition, 4, 46
integers and, 103
multiplication, 4, 49
of sets, 178–180
Cluster points, 176–178
Codomain, 119, 123
Collections, 71. See also Families, of sets
Combinations, 151–157
Combinatorial argument, 152
Combinatorics, 144
Commensurability, 110
Commutative property
addition, 4, 46
binary operations and, 245
multiplication, 4, 49
Commutative rings, 288, 301
Compact sets, uniform continuity and,
239–240
Compactness, 180–183
Comparing real numbers, 5–7
Complement, 3, 63
Complement rule, 147
Completeness, 166
as axiom, 204
LUB, NIP, and, 205–206
of metric space, 204
Complex analysis, 159, 221
Complex conjugate, 336
Complex numbers, 249
as field, 293
Composite integer, 89
Composition, of functions, 131–133, 132f
Conclusion, 19
Conjugates, 277
complex, 336
Conjugation morphisms, 336
Connectedness, of sets, 175
Content, 318, 319
Context, of sets, 3
Continuity, 212, 224–240
implications of, 231–235
IVT and, 231–233
limit v., 225
one-sided, 230–231
and open sets, 233–235
at a point, 224–228
on a set, 228–230
uniform, 235–240
Continuous functions, 224–240
Continuum, real numbers as, 108
Contradiction, proof by, 18, 42, 70, 80–81
Contrapositive, 21, 41, 69
Convergence
of Gauchy sequences, 203–205
of sequences, to infinity, 196–197
of sequences, to real numbers, 190–196
Converse, of if-then statement, 21
Corollary, 50