388 INDEXGreatest lower bound
in a poset, 247
in an ordered field, 3 12, 3 18
of a set of real numbers, 221
of an infinite collection of sets, 128
Group
abelian, 75, 99
cyclic, 75, 99
Group theory, 42Hamel basis, 224
Hamilton, William, 3 19
Hausdorff maximal principle, 224
Heine-Borel theorem, 223,3 18
Hidden quantifier, 89
Hilbert, David, 219
Hyperbolic function, 258Idempotent property
of intersection, 40, 123
of union, 40, 123
Identity
additive, 104, 219, 295, 336, 346, 350
for intersection, 40
for union, 40,121
function, 254, 255
multiplicative, 219, 296, 338, 350
relation, 232
If and only if, 12,61. See also Biconditional
Image
inverse, 252, 270-275
of a function, 253
of an interval under a continuous
function, 317
of a set under a function, 217, 252,
270-275,291
Imaginary number, 321
Imaginary part of a complex number, 14,
32 1
Implication(s), 64, 69
involving predicates in one variable, 97
list of, 70
mathematical significance of, 7 1-72
Implies, 61. See also Conditional
between predicates, 94
Inclusion
mapping, 260,264,280
ordered by, 246
Incomplete ordered field, 313
Increasing
collection of sets, 125, 290
function, 147, 160, 208, 211
Independent variable, 254I
Index, 124
Indexed
by an arbitrary set, 289,290
by N, 124
by the empty set, 292
Indirect proof, 67, 162,204
Induced set functions, 275
Induction
hypothesis, 180
postulate, 331, 341, 343
principle of mathematical, 179,221
principle of transfinite, 224
proof by, 178-189
proofs involving divisibility, 184, 187
proofs involving generalization, 183
proofs involving summation, 181, 186,
188
second principle, 341,342,343
transfinite, 224
Inductive assumption. See Induction
hypothesis
Inductive
definition, 333, 336. See also Recursive
definition
set, 185
Infimum, 247
Infinite collection of sets, 115
greatest lower bound of, 128
least upper bound of, 128
Infinite intersection, 125 - 128
Infinite limit, 131
Infinite set, 5, 6,279,288
countably, 283, 289
Infinite union, 125- 128
Injection, 257, 265. See also Injective
mapping
Injective mapping, 257, 264,267, 271-273
Integer(s), 7, 296, 343-348
even, 59,200,211
modulo m, 297
odd, 59,200,211
positive, 7, 296, 330-343
Integral domain, 114, 347
field of quotients of, 351
Interior point, 191,201
Intermediate value theorem, 3 15
Intersection, 16
arbitrary, 290
associative property of, 3540, 123
commutative property of, 3540, 123
countable, 125, 290
idempotent property of, 40, 123
identity for, 40
infinite, 125-128
of intervals, 161, 168