Cambridge University Press 978-1-108-42418-9 — How to Prove It 3Rd Edition Index More Information

Cambridge University Press 978-1-108-42418-9 — How to Prove It 3rd Edition Index More Information

Index

absorption laws, 21 Chinese remainder theorem, 358 Adleman, Leonard, 360 closed, 259, 263 Alford, W. R., 370 closure, 260, 264, 316 algebraic number, 388 Cocks, Clifford, 360 almost disjoint, 389 Cohen, Paul, 394 amicable numbers, 7 commutative laws antecedent, 45 for ∧ and ∨,21 antisymmetric, 200 for modular arithmetic, 344 arbitrary object, 113 for multiplication, 316 arithmetic mean, 290 compatible, 227, 239, 248 arithmetic mean–geometric mean inequality, complete residue system, 342 290 composite number, 1 associative laws composition, 183, 191, 192, 234 △ for ,45 conclusion, 8, 90 ∧ ∨ for and , 21, 26 conditional, 45 for composition of relations, 186 antecedent of, 45 for modular arithmetic, 344 consequent of, 45 for multiplication, 316 laws, 49 truth table for, 46–47, 49, 148 base case, 273, 304 congruent, 215, 224, 341 Bernstein, Felix, 390 conjecture, 2 biconditional, 54 conjunction, 10 truth table for, 54 big-oh, 239, 301, 303, 314, 388 truth table for, 15 bijection, 246 connective symbol, 10 binary operation, 264 consequent, 45 binary relation, 193 constant function, 238, 247, 249 binomial coefﬁcient, 301, 312 continuum hypothesis, 394 binomial theorem, 302 contradiction, 22, 26, 42 bound variable, 29, 59 laws, 23 bounded quantiﬁer, 72 proof by, 102, 105 contrapositive, 51, 96 Cantor, Georg, 384, 390 law, 51 Cantor’s theorem, 384, 388 converse, 51 Cantor-Schroder-Bernstein¨ theorem, 390 coordinate, 173 cardinality, 372 countable set, 375, 382–389 Carmichael, Robert Daniel, 370 counterexample, 2, 90 Carmichael number, 370 cryptography Cartesian product, 174 public-key, 359–371 cases, proof by, 142 symmetric, 360

455

456 Index De Morgan, Augustus, 21 Fermat number, 289, 340, 369 De Morgan’s laws, 21, 25 Fermat primality test, 369–371 decreasing, 267 Fermat pseudoprime, 369–371 decrypt, 359 Fermat witness, 369–371 denumerable set, 375, 394 Fermat’s little theorem, 357, 369 diagonalization, 386 Fibonacci, 307 difference of sets, 35 Fibonacci numbers, 306, 311, 312, 331 digital signature, 368 finite sequence, 383 directed graph, 194 finite set, 246, 280–283, 289, 292, 322, 372 disjoint, 41 fixed point, 259 pairwise, 161, 216 formula, 12 disjunction, 10 free variable, 29, 59 truth table for, 15 function, 229 disjunctive syllogism, 149 compatible with an equivalence relation, distributive laws 239, 248 for ∩ and ∪, 39, 40 composition of, 234 for ∃ and ∨,77 constant, 238, 247, 249 for ∀ and ∧, 74, 86 domain of, 233 for ∧ and ∨, 21, 25, 39, 40 identity, 230, 251–256 for modular arithmetic, 344 inverse of, 249–259 divides, 126 of two variables, 263 division algorithm, 305, 313, 325, 342 one-to-one, 240 divisor, 324 onto, 240 domain, 183, 191, 233 range of, 233, 242 dominates, 390 restriction of, 237, 247, 258 double negation law, 21, 25 strictly decreasing, 267 dummy variable, 29 strictly increasing, 267 fundamental theorem of arithmetic, 335 edge, 191 element, 28 Godel,¨ Kurt, 394 empty set, 33 geometric mean, 290 encrypt, 359 Gibonacci sequence, 312 equinumerous, 372 given, 93 equivalence class, 216, 217 goal, 93 equivalence relation, 215–228, 239, 381 golden ratio, 315 equivalent formulas, 20 Granville, Andrew, 370 Euclid, 4, 5, 327, 358 graph, 194 Euclidean algorithm, 327, 331 greatest common divisor, 324 extended, 329, 346, 364 greatest lower bound (g.l.b.), 167, 208, 209 least absolute remainder, 332 Euler, Leonhard, 5, 289, 340, 351, 359, 369 harmonic mean, 290 Euler’s phi function, 351, 354 harmonic numbers, 300 Euler’s theorem, 353, 360, 362, 369, 371 Hilbert, David, 340, 394 Euler’s totient function, 351, 354 Hilbert number, 340 even integer, 132, 377 Hilbert prime, 340 exclusive cases, 144 hypothesis, 90 exclusive or, 15, 24 exhaustive cases, 142 idempotent laws, 21 existential instantiation, 120 identity existential quantifier, 58 elements in modular arithmetic, 344, 349 exponentiation in modular arithmetic, 353, function, 230, 251–256 357, 361, 368 relation, 193, 215, 230 efficient computation, 364, 369 iff, 54 image, 230, 233, 268–272 factorial, 5, 165, 294, 296 inclusion-exclusion principle, 381 family of sets, 79 inclusive or, 15 Fermat, Pierre de, 289, 340, 369 increasing, 267

Index 457 index, 78 modus tollens, 108, 113 index set, 78 multiplicative function, 354, 358 indexed family, 78, 79 induction, 273 nand, 25 strong, 304, 311 natural number, 32 induction step, 273 necessary condition, 52 inductive hypothesis, 276, 304 negation, 10 inﬁnite set, 372 truth table for, 15 injection, 240 nor, 24 instance of a theorem, 90 null set, 33 integer, 32 intersection odd integer, 132 of family of sets, 81, 82 one-to-one, 240 of indexed family of sets, 84 one-to-one correspondence, 246 of two sets, 35, 82 onto, 240 interval, 378, 396 ordered pair, 173 inverse ordered triple or quadruple, 179 additive in modular arithmetic, 344, 349 multiplicative in modular arithmetic, 345, pairwise disjoint, 161, 216 349, 351 partial order, 200, 280, 282, 380 of a function, 249–259 strict, 214 of a relation, 183, 191 partition, 216 inverse image, 268–272 Pascal, Blaise, 302 irrational number, 171, 310, 387 Pascal’s triangle, 302 irreﬂexive, 214 perfect number, 5, 358, 359 periodic point, 322 key, 360 pigeonhole principle, 378 public, 360 polynomial, 314 Pomerance, Carl, 370 largest element, 208 power set, 80, 384 least common multiple, 337 premise, 8 least upper bound (l.u.b), 209 preorder, 228, 239, 249 lemma, 219 primality test, 363 Leonardo of Pisa, 307 Fermat, 369–371 limit, 168 Miller-Rabin, 371 linear combination, 328 probabilistic, 363, 369–371 logarithm, 256 prime factorization, 4, 164, 306, 332, loop, 194 335, 363 lower bound, 167, 208 prime number, 1, 75, 78, 163–166, Lucas, Edouard, 313 306, 320 Lucas numbers, 313 largest known, 5 Mersenne, 5 main connective, 17 twin, 6 mathematical induction, see induction proof, 1, 89 maximal element, 208 by cases, 143–147 mean by contradiction, 102, 105 arithmetic, 290 Proof Designer, xi, 128 geometric, 290 proof strategy harmonic, 290 for a given of the form Mersenne, Marin, 5 P → Q, 108 Miller, Gary L., 371 P ↔ Q, 132 Miller-Rabin test, 371 P ∨ Q, 143, 149 Miller-Rabin witness, 371 P ∧ Q, 131 minimal element, 203, 280 ∃xP(x), 120 modular arithmetic, 341–351 ∃! xP(x), 159 modulo, 217 ∀xP(x), 121 modus ponens, 108 ¬P , 105, 108

458 Index proof strategy (cont) Schroder,¨ Ernst, 390 for a goal of the form set, 28, see also countable set; denumerable P → Q, 92, 95, 96 set; empty set (or null set); family of P ↔ Q, 132 sets; finite set; index set; infinite set; P ∨ Q, 145, 147 power set; subset; truth set P ∧ Q, 130 Shamir, Adi, 360 ∃xP(x), 118 σ , 358 ∃! xP(x), 156, 158 -notation, 295 ∀n ∈ N P(n), 273, 304 smallest element, 203 ∀xP(x), 114 strict partial order, 214 ¬P , 101, 102 strict total order, 214 pseudoprime, 369–371 strictly dominates, 390 public key, 360 strong induction, 304, 311 public-key cryptography, 359–371 subset, 41 sufficient condition, 52 quantifier, 58–67 Sun Zi, 358 bounded, 72 surjection, 240 existential, 58 symmetric, 194 negation laws, 68, 70, 73, 135–136 symmetric closure, 214 unique existential, 71, 153–162 symmetric cryptography, 360 universal, 58 symmetric difference, 38, 45, 150–152, 161 quotient, 305, 313, 325–330, 342 τ, 358 Rabin, Michael O., 371 tautology, 22, 26 range, 183, 191, 233, 242 laws, 23 rational number, 32, 171, 377, 393, 396 theorem, 90 real number, 32 total order, 201, 282, 381 rearrangement inequality, 291 strict, 214 recursive transitive, 194 definition, 294 transitive closure, 214, 322 procedure, 288 triangle inequality, 151 refine, 228 truth set, 27, 31, 38, 173, 179 reflexive, 194 truth table, 15–24 relation, 182 truth value, 15 antisymmetric, 200 uncountable set, 375, 382–389 binary, 193 union compatible with an equivalence relation, of family of sets, 81, 82 227 of indexed family of sets, 84 composition of, 183, 191, 192 of two sets, 35, 82 domain of, 183, 191 universal instantiation, 121 identity, 193, 215, 230 universal quantifier, 58 inverse of, 183, 191 universe of discourse, 32 irreflexive, 214 upper bound, 208 range of, 183, 191 reflexive, 194 vacuously true statement, 74 symmetric, 194 valid argument, 9, 18 transitive, 194 variable, 26 relatively prime, 333 bound, 29, 59 remainder, 146, 305, 313, 325–330, dummy, 29 342 free, 29, 59 restriction, 237, 247, 258 Venn diagram, 37, 41 Rivest, Ron, 360 vertex, 191 RSA, 360 rule of inference, 108, 120, 121, well-formed formula, 12 149 well-ordering principle, 309, 327 Russell, Bertrand, 88 Russell’s paradox, 88 Zhang, Yitang, 6