590 INDEX
Logically implies, 69. See also Implication(s)
Lower bound, 246,3 1 1
Lower triangular matrix, 169, 177Mapping(s), 252, 256
codomain of, 256
composition of, 258, 262
equality of, 256
extension of, 265
fixed point of, 332
image of a set under, 252,270-275,291
inclusion, 260, 264, 280
injective, 257,264,267,271-273
into, 266
inverse image of a set under, 252,
270-275
inverse of, 269,274
one-to-one, 257
onto, 252, 257, 266
projection, 276
restriction of, 258, 260
surjective, 266,267, 271, 273
Mathematical generalization, 23 -24, 106
Mathematical induction, 178 - 179. See
also Induction
Mathematical or, 56
Mathematical proof, 115. See also Proof
Matrices
algebraic structure of 2 x 2,297
Matrix, 158
antisymmetric, 158,200
determinant of, 158
diagonal, 99, 169, 177
entries of, 158
inverse of, 200,200
invertible, 191, 193, 200, 220
lower triangular, 169, 177
square, 158, 177
symmetric, 158, 200
transpose of, 158
upper triangular, 99, 169, 177
Maurer, Stephen B., 219 -
Maximal element in a poset, 250
Maximum of two real numbers, 176
Mean value theorem, 24,222
Meet of two elements in a poset, 249, 250
Metric space, 99
Minimal element in a poset, 250
Minimum of two real numbers, 176
Mixed quantifiers, 103
Modulus of a complex number, 14, 322
Modus ponens, 70, 78, 119, 172Modus tollendo ponens, 70
Modus tollens, 70
Multiplication
associative property of, 15, 295,338
cancellation property of, 301, 338, 343
closure property of, 212,295
commutative property of, 15,295, 338,
343
in Q, 349
in R, 356
in Z, 344
of complex numbers, 320
of equivalence classes of Cauchy
sequences, 356
recursive definition of, 336
table, 50
Multiplicative identity, 219, 296, 338, 350
Multiplicative inverse, 220, 296, 338, 350
Mutual inclusion, 1 19
Mutually disjoint
collection of sets, 125, 290, 292
sets, 28
Mutually exclusive cases, 172, 173Natural numbers, 5, 7. See also Positive
integers
Necessary, 72
Negation, 56
as "main connective," 67,73, 97,99, 108
of a predicate, 85
of predicates in several variables, 107
of the five connectives, 67, 68
Negative element in an ordered field, 304
Neighborhood, 191,202
deleted, 201, 202
Nested intervals, 125
Newton, Isaac, 132
Nonempty set, 9
nth power function, 258
nth root function, 258
nth root of complex numbers, 326
Null sequence, 355, 360
Number of elements in a finite set, 43
Number systems, 293-361
complex numbers, 7,296,319-328
construction of, 329-361
natural numbers, 5, 7. See also Positive
integers
positive integers, 7, 296, 330-343
properties of, 293-328
rational numbers, 7, 296, 348-352
real numbers, 7,293-294, 352-361
Number theory, 146, 191