386 INDEXDecreasing collection of sets, 125, 290
Dedekind, Richard, 330
Dedekind cuts, 329
Definitionfs), 1 15
inductive, 333, 336. See also Recursive
definition
informal, 4
recursive, 302,309,330,333, 335, 336, 342
role of, in proof-writing, 196
Deleted neighborhood, 201
Delta band, 132
DeMoivre's theorem, 324, 326
DeMorgan's laws
generalized, 127
logic, 68
sets, 34, 41, 123, 152
Denial, 56. See also Negation
Denumerable set, 283
Dependence
in logic, 105, 106, 109, 191-193
linear, 165,212
Dependent variable, 254
Derivative, 24, 99, 101, 131, 141, 178, 180
Description method, 5
Detachment, 70
Determinant of a 2 x 2 matrix, 158,220
Diagonal matrix, 99,169,177
Difference
in a field, 301
set theoretic, 19, 25
Differentiable function, 71, 74, 101
Differential equations, 222
Direct product of a collection of sets, 291
Direct proof, 162
Discovery, 3, 164
Disjoint sets, 17, 28
Disjunction, 56
associative property of, 68
between predicates, 85
commutative of, 68
law of, 70
proofs of conclusions involving, 204-206
Distance, in an ordered field, 308
Distributive property
generalized, 28, 183
of addition over multiplication
(invalidity of), 38
of conjunction over disjunction, 69
of disjunction over conjunction, 69
of intersection over symmetric difference,
41, 152, 159
of intersection over union, 32,41, 120
of multiplication over addition, 15, 321,
296, 337
of union over intersection, 32,41
Divides, 109, 147, 191, 193, 199, 210, 211,
23 1,246,249
Divisibility
induction proofs involving, 184, 187
ordered by, 231,250
Division algorithm for 2, 218, 237, 239, 244
quotient, 239
remainder, 239
Division into cases, 119, 167, 170, 172-177
Domain
of a function, 253
of a relation, 233
Domain of discourse, 83
Dummy variable, 88,125
Dynamieview
of a function, 254
of a relation, 231Element, 4
Elementary methods of proof, 146-189
Elementhood method, 117. See also
Choose method
Empty set, 9, 36-37
as a universal set, 100
as an indexing set, 292
is inductive, 187
proofs involving, 120-122
English language translations
of logical connectives, 72
of quantified predicates, 89-93, 108, 110
Entries of a matrix, 158
Epsilon band, 132
Epsilon-delta definition of limit, 52, 1 15,
132
meaning of, 132-140
negation of, 133
Epsilon-delta proof(s), 196- 199, 202
negative, 139
specialization in, 198
Equality
as an equivalence relation, 227
of complex numbers, 320
of functions, 253, 264
of mappings, 256
of ordered pairs, 21, 123,229
Equality of sets, 10, 15, 116
reflexive property of, 10,40
symmetric property of, 10,40
transitive property of, 10,40, 211
Equivalence@), 64
involving predicates in one variable, 96
list of, 68
mathematical significance of, 66-67