392 INDEX
Predicate(s) (continued)
logical equivalence between, 83,94
logical implication between, 94
Premise
of a conditional statement form, 61
of an argument, 76
partial, 76
Preserving an operation, 346
Prime integer, 74, 2 1 7
Principal value of the argument of a
complex number, 324
Principle
pigeon hole, 223
of inductive definition, 336
of mathematical induction, 179
of transfinite induction, 224
second induction, 341, 342, 343
well-ordering, 187, 218, 221, 330, 340,
341,343
Projection mappings, 276
Proof, 149
advanced methods, 190-224
by choose method, 1 17, 162
by contradiction, 121, 204, 209, 210
by contrapositive, 67, 204, 205
by division into cases, 119, 170, 172-177
by enumerating cases, 154
by example, 154
by induction, 178- 189
by mutual inclusion, 119
by specialization, 122, 167, 170, 171-172,
174-177, 179, 198,220
by transitivity, 148, 150
by truth table, 64
by Venn diagram, 30
direct, 162
elementary methods, 146- 189
epsilon-delta, 196- 199, 202
final form of, 162, 164, 197
in set theory, 115-124
indirect, 67, 162, 204
involving the empty set, 120-122
mathematical, 115
of conclusion involving disjunction,
204-206
of equivalence, 166
of existence, 212, 216-222
of uniqueness, 21 2, 214-216,219-222
Proper subset, 12, 15, 165, 169
Proposition, 53. See also Statement
Propositional calculus, 53-80
theorems of, 64-76
Propositional function, 52. See also
Predicate
Purely imaginary number, 32 1
Quantification, 88-95
Quantifier@), 53, 82
existential, 88
hidden, 89
mixed, 103
universal, 88
Quotient(s), 239
field of, 351
in a field, 301
Quotient structure, 244Range
of a function, 253
of a relation, 233
Rational function, 254, 264
Rational(s), 7, 110
construction of, 348-352
field of, 296
number(s), 191, 199, 21 1
Real number(s), 7, 110
construction of, 352-361
familiarity of, 293
field of, 294, 296
ordering of, 358
Real part of a complex number, 14, 321
Recursive definition, 302, 309, 330, 333,
335,336,342
of addition, 335
of factorial, 336, 342
of Fibonacci number, 336, 342
of multiplication, 336
of nth power, 336, 342
Reductio ad absurdum, 70, 204
Reflexive
in definition of equivalence relation, 236
in definition of partial ordering, 245
property of a relation, 232
property of logical equivalence, 68
property of set equality, 10, 40
property of the subset relation, ll,40
Relation(s), 227-292
antisymmetric, 232
between sets, 228, 230
congruence modulo, 232, 237,239, 243
divides, 231, 246, 249
domain of, 233
dynamic view of, 23 1
equivalence, 227, 235, 236
identity, 232
inverse of, 233
irreflexive, 309
less than, 232
less than or equal to, 231, 245, 246
on a set, 230