INDEX in Java

Create UPCA in Java INDEX

Partitioning, 54 Pascal, Blaise, 127 Pascal s triangle, 127 130 Peano, Giuseppe, 68 Peano s axioms, 68 71, 237 Percentages, probability and, 124 Permutations, 108 110 Physics, 3 complex numbers and, 52 group theory and, 196 matrices and, 135 Physiology, Alan Turing and, 203 Pigeonhole principle, 105 108 Pinching theorem, for sequences, 253 Plaintext, 204 Polynomial functions, 63 Positive integers See Natural numbers Power sets, 48 Predicate calculus, 2 Prime factors, 185 Prime numbers natural numbers and, 185 RSA encryption and, 227 Probability, 38, 124 127 Problems coin tosses, 124, 129, 153 coloring, 172 178 hospital alarm system, 239 241 K nigsberg bridge, 169 172 traveling salesman, 178 180 Proof, 3, 42 by contradiction, 29 32 direct, 24 29 by induction, 32 37, 48 methods of, 23 40 types of, 24 Propositional calculus, 1 Propositional logic, 235 Propositions, 1, 25, 42 p-series, convergence of, 279 282, 286, 291 Public key encryption, 232 Pythagoras, 31 Pythagorean theorem, 28, 30 Ramsey, Frank Plumpton, 130 Ramsey theory, 39, 130 132 Range of a function, 57 of a relation, 53 Rational fractions, 266 Rational numbers (Q), 29 31, 52 building, 79 86 real numbers and, 86 89, 91 94 Ratio test, 291 294 Real numbers (R), 52, 86 94 complex numbers and, 96 99 construction of, 91 94 in nite sequences of, 244 250 rational numbers and, 86 89 special sequences and, 254 Recursions, 121 124 Regions (faces), 165 Relations, 53 55, 95, 191 Relatively prime integers, 184, 225, 228 Remainder (r ), 184 Repeating decimals, 266 268 Riemann, Bernhard, 202 Riemann hypothesis, 202 Riemann zeta function, 202 Ringel, Gerhard, 176 Rivest, R, 222 Robinson, Abraham, 94 Root test, 294 298 Rows of matrices, 136 RSA encryption, 221 233 explanation of, 228 implementing, 226 228 preparation for, 222 226 Rules functions and, 56 of logic, 1 22, 24

Schedules, linear programming and, 156 Science, 3, 118 Second-order logic, 19 Sentential logic, 2 Sequences, 243 256 basic theorem for, 250 253 pinching theorem for, 253 properties of, 250 of real numbers, 244 250 vs series, 261 special sequences and, 254 tail of, 250 Series, 257 299 basic properties of, 270 273 condensed, 280 convergence and, 261

Q (quotient), 52 See also Rational numbers Quadratic coef cient, 63 Quanti ers, 2, 16 19 Quantum mechanics, group theory and, 196 Quaternions, 101 Quotient (Q), 52 See also Rational numbers

R See Real numbers r (remainder), 184

Top-order coef cient, 63 Torus, 175, 177 Transitivity, 54 55 Transliteration, cryptography and, 207, 213, 216, 227 Traveling salesman problem (TSP), 178 180 Tree theory, 180 Triangle inequality, 90 Trigraphs, 204 True/false, 2, 29 Truth, 2 Truth tables, 4, 8 TSP (traveling salesman problem), 178 180 Turing, Alan Mathison, 199, 202 Turing machine, 200 202

divergence and, 261, 263, 288 291 examples of, 260 263, 268 geometric, 265, 273 279 harmonic, 263 265, 280 vs sequences, 261 Series of powers, 265, 273 279 Setbuilder notation, 42 Set formulation of induction, 36 Set of integers, 52 Sets, 41 50 boolean algebra and, 237 elements of set theory and, 42 46 empty, 44 power, 48 of rational numbers, 79 universal, 45 Venn diagrams and, 46 Set-theoretic difference, 44 Set-theoretic isomorphism, 63 Set-theoretic product, 47 48 Set-theoretic relationships, 46 Set unions, 44 Seymour, Paul, 178 Shamir, A, 222 Shift transformations, 207 209 Sigma ( ), 113, 117 Social sciences, matrices and, 135 Solutions/answers to exercises, 325 346 Spheres graphs and, 165, 169 torus and, 174 Statements, 1, 6 Statistics, matrices and, 135 Stirling s formula, 179 Stone representation theorem, 236 Subgroups, 190 Subtraction, 52, 75, 81 Sum, 113, 116 Supremums, 87 90, 93 Surjections, 58 Sylow s theorems, 194

Ultra lters, 94 Unde nables, 41 Union of sets, 44 Unions, Venn diagrams and, 46 Univalent mapping, 58 Universal sets, 45 Universe, 17 Upper bounds, 86 89, 92

Venn diagrams, 46 Vertices, 165, 169 172 four-color theorem and, 223 odd/even, 171 von Neumann, John, 199

Weather, Markov chains and, 153 156

Yanghui, 127 Yanghui s triangle, 127 Youngs, J W T, 176

Task management, linear programming and, 156 Terminating decimal expansions, 52 Theorems, 25, 42 Dirichlet, 31 Euclid s, 185 Pythagorean, 28, 30 there exists ( ), 2, 16, 17 19 Third-order logic, 19

z, of modulus, 100 Zahlen, 52 Zermelo, Ernst, 95f Zero, negative numbers and, 73 Zero knowledge, 229 233 Zorn s lemma, 95