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Velleman D.J. — How to Prove It: A Structured Approach
Velleman D.J. — How to Prove It: A Structured Approach

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Название: How to Prove It: A Structured Approach

Автор: Velleman D.J.

Аннотация:

Many students have trouble the first time they take a mathematics course in which proofs play a sigficant role. This book will prepare students for such courses by teaching them techniques for writing and reading proofs. No background beyond high school mathematics is assumed. The book begins with logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. This understanding of the language of mathematics serves as the basis for a detailed discussion of the most important techniques used in proofs, when and how to use them, and how they are combined to produce comples proofs. Material on the natural numbers, relations, functions, and infinite sets provides practice in writing and reading proofs, as well as supplying background that will be valuable in most theoretical mathematics courses.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2006

Количество страниц: 336

Добавлена в каталог: 23.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\Sigma$-notation      281—282
Absorption laws      21
Antecedent      44
antisymmetric      189
Arbitrary object      108
Arithmetic mean      276
Arithmetic-geometric mean inequality      276
Associative laws      22 23
Associative laws for $\wedge$ and $\vee$      21 25
asymmetric      210
Bernstein, Felix      322
Biconditional      23 52 53
Biconditional, truth table for      23 52
Big-Oh      235
Bijecuon      182 242
Binary relation      182 242
Binomial coefficient      288
Binomial theorem      260 288
Bound variable      29
Bounded quantifier      57 68
Canior — Schroeder — Bernstein Theorem      322—327
Canter's Theorem      318 320 321
Cantor, Georg      318
Cardinality      307
Cartezian product      163—171
cases      136
Closure of a set under a function      303
Closure, reflexive      202
Closure, reflexive symmetric      212
Closure, symmetric      204
Closure, symmetric transitive      212
Closure, transitive      204 209 300
Cohen, Paul      327
Commutative laws for $\wedge$ and $\vee$      21 23 52
Compatible      225 236
composition      177 178 182 231 300
Conclusion      8 85
Conditional laws      44—45 47 50
Conditional, antecedent of      44
Conditional, consequent of      44
Conditional, truth table for      44—45 47
Congruent      213
Conjecture      2
Conjunction      10
Connective symbol      10
Consequent      44
Constant function      235 244
Continuum Hypothesis      326—327
Contradiction      22 23 26 32 41
Contradiction law      23
Contradiction, proof by      96 97 98 99
Contrapositive      49 91
Contrapositive law      49
CONVERSE      49
Coordinate      163
Countable set      310
Counterexample      2 85
DeMorgan's law      20 21 22 23 25 39 47 50
Denumerable set      318 326
Diagonalization      320
Difference of sets      34
Directed graph      183
Disjoint      40
Disjoint pairwise      153 214
Disjunction      10
Disjunctive syllogism      142
Distributive laws      38—39
Distributive laws for $\cap$ and $\cup$      38—39
Distributive laws for $\exists$      70
Distributive laws for $\forall$ and $\vee$      70
Distributive laws for $\wedge$ and $\vee$      21 23
divides      121
Division algorithm      290 291
Domain      172 230
Double negation law      21 22 23: 47
EDGE      181
Element      27
Empty set      32
Equintimerous sets      306—312
Equivalence class      214 215
Equivalence relation      213—222 309
Equivalent formulas      19 21
Even integer      134 312
Exclusive cases      138
Exclusive OR      15 24
Exhaustive cases      136
Existential instantiation      115
Existential quantifier      55 58
Factorial      282
Family of sets      75
Fermat numbers      276
Fermat, Pierre de      276
fibonacci      291
Fibonacci numbers      291—293 297
Finite sequence      317 321—322
Finite set      306
Fixed point      255
Formula      12
Free variable      28 55
Function, compatible with an equivalence relation      225
Function, composition of      232—233
Function, domain of      230
Function, function of      229
Function, identity      227
Function, inverse of      245—252
Function, one-to-one      236
Function, onto      236
Function, range of      239
Function, restriction of      234 243
Geometric mean      357
Gihonacci sequence      297
givens      88
Goal      88
Goedel, Kurt      327
Graph      183
Greatest common divisor      299
Greatest lower bound (g.l.b.)      197
Harmonic mean      277
Harmonic numbers      286
Hilbert, David      326
Hypothesis      85
Idempolent laws      21 22
Identity relation      183 203—204 213
IFF      52
Image      227 230 255—259
Inclusion-Exclusion Principle      369
Inclusive or      15
INDEX      74
Index set      74
Indexed family      74 75
Induction step      260 270
Induction, strong      288—295
Inductive hypothesis      263 268 272
Inference, rule of      107
Infinite sets      306
Injections      236
Instance of a theorem      85
Integers      31 32
Intersection of family of sets      77 78
Intersection of indexed family of sets      79
intersection of two sets      34
interval      31 328
Inverse      173 245—252
Inverse image      255—259
Irrational number      161—162
Irrefiexive      204
largest element      197
Least upper bound      197
Lemma      217
LIMIT      161
Logarithm      252
Loop      184
Lower bound      197
Lucas numbers      297
Lucas, Edouard      297
Main connective      17
Mathematical induction      See induction
Maximal element      197
Mersenne, Marin      5
Minimal element      192 267—269
MODULO      215 221
modus ponens      103
Modus Tollens      103 104 105 107
NAND      24
Natural number      31
Necetttiuy condition      50
Negation      10
NOR      24
Null set      32
Odd integer      24 133 134
One-to-one      236
One-to-one correspondence      242
Onto      236
Ordered pair      163—171
Ordered triple or quadruple      169
Pairwise disjoint      153 214
Partial order      190 254 267 268—269 314
Partial order, strict      211
Partition      214 215
Pascal's triangle      288
Pastil, Blaise      288
Perfect number      5
Pigeonhole Principle      313
Polynomial      299
Power set      75—76 119 318
Premise      8
Preorder      225
Prime number      1 74 94 156—158
Prime number, largest known      5
Prime number, Mersenne      5
Prime number, twint      6
proof      1 84
Proof by cases      137
Proof by contradiction      96 97 98 99
Proof Designer      102 373—374
Proof Designer, Preface and Appendix      373—374
Proof strategy for a given of the form, $P \rightarrow Q$      102
Proof strategy for a given of the form, $P \rightleftarrow Q$      126
Proof strategy for a given of the form, $P \vee Q$      142—143
Proof strategy for a given of the form, $P \wedge Q$      124
Proof strategy for a given of the form, $\exists ! x P(x)$      152
Proof strategy for a given of the form, $\exists x P(x)$      115
Proof strategy for a given of the form, $\forall x P(x)$      115
Proof strategy for a given of the form, $\neg P$      99—100 102
Proof strategy for a goal of the form, $P \rightarrow Q$      88 90 91
Proof strategy for a goal of the form, $P \rightleftarrow Q$      126
Proof strategy for a goal of the form, $P \vee Q$      138 141—142
Proof strategy for a goal of the form, $P \wedge Q$      124
Proof strategy for a goal of the form, $\exists x P(x)$, $\exists ! x P(x)$      149
Proof strategy for a goal of the form, $\forall x P(x)$, $\forall n \in \mathbb{N} P(n)$      260 289
Proof strategy for a goal of the form, $\neg P$      95 96
Proper subset      203
Quantifier      55—64
Quantifier, bounded      68 57
Quantifier, existential      58
Quantifier, negation laws      65 66 69
Quantifier, unique existential      146
Quantifier, universal      55
Quotient      290
RANGE      173 239
Rational number      31 311 326 328
Real number      31
Recursive definition      280
Recursive procedure      274
Refine      225
Reflexive      184
Reflexive closure      202
Reflexive symmetric closure      212
Relation, antisymmetric      212
Relation, asymmetric      210 346
Relation, binary      182 242
Relation, composition of      176 180—181
Relation, domain of      173
Relation, identity      183 203—204 213
Relation, inverse of      173
Relation, irreflexive      204
Relation, range of      311
Relation, reflexive      184
Relation, symmetric      184
Relation, transitive      184
Remainder      290 139—140
Restriction      234 243
Rule of inference      107
Russell's paradox      83
Russell, Bertrand      83
Schroeder, Emst      322
SEQUENCE      297—298
Set      27. See also countable set; denumerable set; empty set (or null set); family of sets; finite set; index set infinite
Smallest element      184
Strict partial order      204
Strict total order      204
Strictly dominates      322
Strong induction      288—295
Subset      40 184
Subset, proper      203
Sufficient condition      50
Surjection      236
Symmetric closure      204 205
Symmetric difference      36 43 154
Symmetric transitive closure      40 212 184
Tautology      23
Tautology laws      23
Theorem      85
Total order      190 269—270 275
Total order, strict      204
Transitive closure      204
Trichotomy      204
Truth set      27 30 37 163
Truth table      14—23
Truth value      14
Uncountable set      315—321
Union of a family of sets      78
Union of an indexed family of sets      79
union of two sets      78
Universal instantiation      15
Universal quantifier      55
Universe of discourse      31 328
Upper bound      197
Vacuously true statement      70
Valid argument      9
Variable      26
Variable, bound      29
Variable, free      28 55
Venn digram      36 40—41
Vertex      182
Well-formed formula      12
Well-ordered principle      294 295 299
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