Bridge to Abstract Mathematics: Mathematical Proof and Structures

384 INDEX

Associative property (continued)

of multiplication, 15, 295, 338

of subtraction (invalidity of), 154

of symmetric difference, 41, 158

of union, 32, 40, 123

Axiom, 149,298

additive identity, 219.

additive inverse, 220

based on Peano's Postulates, 331

least upper bound, 218,221

multiplicative identity, 219

multiplicative inverse, 220

of choice, 224,288, 291

Band
delta, 132
epsilon, 132
Biconditional
between predicates, 85
connective, 61
English language translation of the, 72
law, 69
stronger than the conditional, 70
Bijection, 269, 273, 274
Binary operation, 16, 294
Binomial coefficient, 50, 156, 157
Binomial theorem, 188
Boole, George, 53
Boolean algebra, 53
Bound
lower, 246, 31 1
upper, 246,3 1 1
Bounded
above, 21 7,220,246, 3 1 1
below, 220,246, 3 1 1
function, 202
interval, 8
set, 21 1,220
subset of a metric space, 99
subset of a poset, 246
subset of an ordered field, 3 11
Brouwer, L. E. J., 219

Calculus, 24, 132

advanced, 146, 191

elementary, 128

predicate, 53, 81-1 14

propositional, 53-80

Cancellation

additive, 299, 336, 342

in composition of functions, 263, 266,

267,268

multiplicative, 301, 338, 343

Cantor, Georg, 3,277, 280,281,287

Cantor's theorem, 284

Cardinal number(s), 277

comparability of, 288

less than, between, 284

less than or equal to, between, 284

of the continuum, 283

same, 278

Cardinality, 127, 252, 277-289

Cartesian product, 21, 22, 228, 229

Cauchy sequence(s), 203, 329, 330, 353, 354

equivalence classes of, 356

equivalence of, 356

Cells of a partition, 240

Chain in a poset, 248

Chain rule, 260

Choice

axiom of, 224, 288, 291

function, 29 1

Choose method, 117

Circular reasoning, 86

Classification theorem, 222

Closed

form of a summation formula, 181

interval, 8

subset of a metric space, 99

set, 201, 211, 212

Closure

of addition, 212, 295

of multiplication, 212, 295

of subtraction, 347

Cluster point of a sequence, 203, 220

Condomain of a mapping, 256

Cohen, Paul, 287

Collection of sets

arbitrary, 127, 252, 289, 290

arbitrary intersection of, 290

arbitrary union of, 290

countably infinite, 127, 290

decreasing, 125, 290

direct product of, 291

increasing, 125, 290

indexed by an arbitrary set, 290

indexed by N, 124

infinite, 1 15

infinite (countably) intersection of, 125

infinite (countably) union of, 125

mutually disjoint, 125, 290, 292

pairwise disjoint, see Mutually disjoint

sets

College Mathematics Journal, 219

Combinations, 47

Commutative property

of addition, 15, 295, 336