{"product_id":"a-first-course-in-mathematical-logic-and-set-theory-isbn-9780470905883","title":"A First Course in Mathematical Logic and Set Theory","description":"\u003cp\u003e\u003cb\u003eA mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eHighlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, \u003ci\u003eA First Course in Mathematical Logic and Set\u003c\/i\u003e \u003ci\u003eTheory \u003c\/i\u003eintroduces how logic is used to prepare and structure proofs and solve more complex problems.\u003c\/p\u003e \u003cp\u003eThe book begins with propositional logic, including two-column proofs and truth table applications, followed by first-order logic, which provides the structure for writing mathematical proofs. Set theory is then introduced and serves as the basis for defining relations, functions, numbers, mathematical induction, ordinals, and cardinals. The book concludes with a primer on basic model theory with applications to abstract algebra. \u003ci\u003eA First Course in Mathematical Logic and Set Theory \u003c\/i\u003ealso includes:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eSection exercises designed to show the interactions between topics and reinforce the presented ideas and concepts\u003c\/li\u003e \u003cli\u003eNumerous examples that illustrate theorems and employ basic concepts such as Euclid’s lemma, the Fibonacci sequence, and unique factorization\u003c\/li\u003e \u003cli\u003eCoverage of important theorems including the well-ordering theorem, completeness theorem, compactness theorem, as well as the theorems of Löwenheim–Skolem, Burali-Forti, Hartogs, Cantor–Schröder–Bernstein, and König\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eAn excellent textbook for students studying the foundations of mathematics and mathematical proofs, \u003ci\u003eA First Course in Mathematical Logic and Set Theory \u003c\/i\u003eis also appropriate for readers preparing for careers in mathematics education or computer science. In addition, the book is ideal for introductory courses on mathematical logic and\/or set theory and appropriate for upper-undergraduate transition courses with rigorous mathematical reasoning involving algebra, number theory, or analysis.\u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eAcknowledgments xv\u003c\/p\u003e \u003cp\u003eList of Symbols xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Propositional Logic 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Symbolic Logic 1\u003c\/p\u003e \u003cp\u003ePropositions 2\u003c\/p\u003e \u003cp\u003ePropositional Forms 5\u003c\/p\u003e \u003cp\u003eInterpreting Propositional Forms 7\u003c\/p\u003e \u003cp\u003eValuations and Truth Tables 10\u003c\/p\u003e \u003cp\u003e1.2 Inference 19\u003c\/p\u003e \u003cp\u003eSemantics 21\u003c\/p\u003e \u003cp\u003eSyntactics 23\u003c\/p\u003e \u003cp\u003e1.3 Replacement 31\u003c\/p\u003e \u003cp\u003eSemantics 31\u003c\/p\u003e \u003cp\u003eSyntactics 34\u003c\/p\u003e \u003cp\u003e1.4 Proof Methods 40\u003c\/p\u003e \u003cp\u003eDeduction Theorem 40\u003c\/p\u003e \u003cp\u003eDirect Proof 44\u003c\/p\u003e \u003cp\u003eIndirect Proof 47\u003c\/p\u003e \u003cp\u003e1.5 The Three Properties 51\u003c\/p\u003e \u003cp\u003eConsistency 51\u003c\/p\u003e \u003cp\u003eSoundness 55\u003c\/p\u003e \u003cp\u003eCompleteness 58\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 First-Order Logic 63\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Languages 63\u003c\/p\u003e \u003cp\u003ePredicates 63\u003c\/p\u003e \u003cp\u003eAlphabets 67\u003c\/p\u003e \u003cp\u003eTerms 70\u003c\/p\u003e \u003cp\u003eFormulas 71\u003c\/p\u003e \u003cp\u003e2.2 Substitution 75\u003c\/p\u003e \u003cp\u003eTerms 75\u003c\/p\u003e \u003cp\u003eFree Variables 76\u003c\/p\u003e \u003cp\u003eFormulas 78\u003c\/p\u003e \u003cp\u003e2.3 Syntactics 85\u003c\/p\u003e \u003cp\u003eQuantifier Negation 85\u003c\/p\u003e \u003cp\u003eProofs with Universal Formulas 87\u003c\/p\u003e \u003cp\u003eProofs with Existential Formulas 90\u003c\/p\u003e \u003cp\u003e2.4 Proof Methods 96\u003c\/p\u003e \u003cp\u003eUniversal Proofs 97\u003c\/p\u003e \u003cp\u003eExistential Proofs 99\u003c\/p\u003e \u003cp\u003eMultiple Quantifiers 100\u003c\/p\u003e \u003cp\u003eCounterexamples 102\u003c\/p\u003e \u003cp\u003eDirect Proof 103\u003c\/p\u003e \u003cp\u003eExistence and Uniqueness 104\u003c\/p\u003e \u003cp\u003eIndirect Proof 105\u003c\/p\u003e \u003cp\u003eBiconditional Proof 107\u003c\/p\u003e \u003cp\u003eProof of Disunctions 111\u003c\/p\u003e \u003cp\u003eProof by Cases 112\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Set Theory 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Sets and Elements 117\u003c\/p\u003e \u003cp\u003eRosters 118\u003c\/p\u003e \u003cp\u003eFamous Sets 119\u003c\/p\u003e \u003cp\u003eAbstraction 121\u003c\/p\u003e \u003cp\u003e3.2 Set Operations 126\u003c\/p\u003e \u003cp\u003eUnion and Intersection 126\u003c\/p\u003e \u003cp\u003eSet Difference 127\u003c\/p\u003e \u003cp\u003eCartesian Products 130\u003c\/p\u003e \u003cp\u003eOrder of Operations 132\u003c\/p\u003e \u003cp\u003e3.3 Sets within Sets 135\u003c\/p\u003e \u003cp\u003eSubsets 135\u003c\/p\u003e \u003cp\u003eEquality 137\u003c\/p\u003e \u003cp\u003e3.4 Families of Sets 148\u003c\/p\u003e \u003cp\u003ePower Set 151\u003c\/p\u003e \u003cp\u003eUnion and Intersection 151\u003c\/p\u003e \u003cp\u003eDisjoint and Pairwise Disjoint 155\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Relations and Functions 161\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Relations 161\u003c\/p\u003e \u003cp\u003eComposition 163\u003c\/p\u003e \u003cp\u003eInverses 165\u003c\/p\u003e \u003cp\u003e4.2 Equivalence Relations 168\u003c\/p\u003e \u003cp\u003eEquivalence Classes 171\u003c\/p\u003e \u003cp\u003ePartitions 172\u003c\/p\u003e \u003cp\u003e4.3 Partial Orders 177\u003c\/p\u003e \u003cp\u003eBounds 180\u003c\/p\u003e \u003cp\u003eComparable and Compatible Elements 181\u003c\/p\u003e \u003cp\u003eWell-Ordered\u003c\/p\u003e \u003cp\u003eSets 183\u003c\/p\u003e \u003cp\u003e4.4 Functions 189\u003c\/p\u003e \u003cp\u003eEquality 194\u003c\/p\u003e \u003cp\u003eComposition 195\u003c\/p\u003e \u003cp\u003eRestrictions and Extensions 196\u003c\/p\u003e \u003cp\u003eBinary Operations 197\u003c\/p\u003e \u003cp\u003e4.5 Injections and Surjections 203\u003c\/p\u003e \u003cp\u003eInjections 205\u003c\/p\u003e \u003cp\u003eSurjections 208\u003c\/p\u003e \u003cp\u003eBijections 211\u003c\/p\u003e \u003cp\u003eOrder Isomorphims 212\u003c\/p\u003e \u003cp\u003e4.6 Images and Inverse Images 216\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Axiomatic Set Theory 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Axioms 225\u003c\/p\u003e \u003cp\u003eEquality Axioms 226\u003c\/p\u003e \u003cp\u003eExistence and Uniqueness Axioms 227\u003c\/p\u003e \u003cp\u003eConstruction Axioms 228\u003c\/p\u003e \u003cp\u003eReplacement Axioms 229\u003c\/p\u003e \u003cp\u003eAxiom of Choice 230\u003c\/p\u003e \u003cp\u003eAxiom of Regularity 234\u003c\/p\u003e \u003cp\u003e5.2 Natural Numbers 237\u003c\/p\u003e \u003cp\u003eOrder 239\u003c\/p\u003e \u003cp\u003eRecursion 242\u003c\/p\u003e \u003cp\u003eArithmetic 243\u003c\/p\u003e \u003cp\u003e5.3 Integers and Rational Numbers 249\u003c\/p\u003e \u003cp\u003eIntegers 250\u003c\/p\u003e \u003cp\u003eRational Numbers 253\u003c\/p\u003e \u003cp\u003eActual Numbers 256\u003c\/p\u003e \u003cp\u003e5.4 Mathematical Induction 257\u003c\/p\u003e \u003cp\u003eCombinatorics 260\u003c\/p\u003e \u003cp\u003eEuclid’s Lemma 264\u003c\/p\u003e \u003cp\u003e5.5 Strong Induction 268\u003c\/p\u003e \u003cp\u003eFibonacci Sequence 268\u003c\/p\u003e \u003cp\u003eUnique Factorization 271\u003c\/p\u003e \u003cp\u003e5.6 Real Numbers 274\u003c\/p\u003e \u003cp\u003eDedekind Cuts 275\u003c\/p\u003e \u003cp\u003eArithmetic 278\u003c\/p\u003e \u003cp\u003eComplex Numbers 280\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Ordinals and Cardinals 283\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Ordinal Numbers 283\u003c\/p\u003e \u003cp\u003eOrdinals 286\u003c\/p\u003e \u003cp\u003eClassification 290\u003c\/p\u003e \u003cp\u003eBuraliForti and Hartogs 292\u003c\/p\u003e \u003cp\u003eTransfinite Recursion 293\u003c\/p\u003e \u003cp\u003e6.2 Equinumerosity 298\u003c\/p\u003e \u003cp\u003eOrder 300\u003c\/p\u003e \u003cp\u003eDiagonalization 303\u003c\/p\u003e \u003cp\u003e6.3 Cardinal Numbers 307\u003c\/p\u003e \u003cp\u003eFinite Sets 308\u003c\/p\u003e \u003cp\u003eCountable Sets 310\u003c\/p\u003e \u003cp\u003eAlephs 313\u003c\/p\u003e \u003cp\u003e6.4 Arithmetic 316\u003c\/p\u003e \u003cp\u003eOrdinals 316\u003c\/p\u003e \u003cp\u003eCardinals 322\u003c\/p\u003e \u003cp\u003e6.5 Large Cardinals 327\u003c\/p\u003e \u003cp\u003eRegular and Singular Cardinals 328\u003c\/p\u003e \u003cp\u003eInaccessible Cardinals 331\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Models 333\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 First-Order Semantics 333\u003c\/p\u003e \u003cp\u003eSatisfaction 335\u003c\/p\u003e \u003cp\u003eGroups 340\u003c\/p\u003e \u003cp\u003eConsequence 346\u003c\/p\u003e \u003cp\u003eCoincidence 348\u003c\/p\u003e \u003cp\u003eRings 353\u003c\/p\u003e \u003cp\u003e7.2 Substructures 361\u003c\/p\u003e \u003cp\u003eSubgroups 363\u003c\/p\u003e \u003cp\u003eSubrings 366\u003c\/p\u003e \u003cp\u003eIdeals 368\u003c\/p\u003e \u003cp\u003e7.3 Homomorphisms 374\u003c\/p\u003e \u003cp\u003eIsomorphisms 380\u003c\/p\u003e \u003cp\u003eElementary Equivalence 384\u003c\/p\u003e \u003cp\u003eElementary Substructures 388\u003c\/p\u003e \u003cp\u003e7.4 The Three Properties Revisited 394\u003c\/p\u003e \u003cp\u003eConsistency 394\u003c\/p\u003e \u003cp\u003eSoundness 397\u003c\/p\u003e \u003cp\u003eCompleteness 399\u003c\/p\u003e \u003cp\u003e7.5 Models of Different Cardinalities 409\u003c\/p\u003e \u003cp\u003ePeano Arithmetic 410\u003c\/p\u003e \u003cp\u003eCompactness Theorem 414\u003c\/p\u003e \u003cp\u003eLöwenheim–Skolem Theorems 415\u003c\/p\u003e \u003cp\u003eThe von Neumann Hierarchy 417\u003c\/p\u003e \u003cp\u003eAppendix: Alphabets 427\u003c\/p\u003e \u003cp\u003eReferences 429\u003c\/p\u003e \u003cp\u003eIndex 435\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eMichael L. O'Leary, PhD,\u003c\/b\u003e is Professor of Mathematics at the College of DuPage in Glen Ellyn, Illinois. He received his doctoral degree in mathematics from the University of California, Irvine in 1994 and is the author of \u003ci\u003eRevolutions of Geometry\u003c\/i\u003e, also published by Wiley.   \u003c\/p\u003e\u003cp\u003e\u003cb\u003eA mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003eHighlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, \u003ci\u003eA First Course in Mathematical Logic and Set Theory\u003c\/i\u003e introduces how logic is used to prepare and structure proofs and solve more complex problems. \u003c\/p\u003e\u003cp\u003eThe book begins with propositional logic, including two-column proofs and truth table applications, followed by first-order logic, which provides the structure for writing mathematical proofs. Set theory is then introduced and serves as the basis for defining relations, functions, numbers, mathematical induction, ordinals, and cardinals. The book concludes with a primer on basic model theory with applications to abstract algebra. \u003ci\u003eA First Course in Mathematical Logic and Set Theory\u003c\/i\u003e also includes: \u003c\/p\u003e\u003cul\u003e \u003cli\u003eSection exercises designed to show the interactions between topics andreinforce the presented ideas and concepts\u003c\/li\u003e \u003cli\u003eNumerous examples that illustrate theorems and employ basic conceptssuch as Euclid's lemma, the Fibonacci sequence, and unique factorization\u003c\/li\u003e \u003cli\u003eCoverage of important theorems including the well-ordering theorem,completeness theorem, compactness theorem, as well as the theorems ofLöwenheim–Skolem, Burali-Forti, Hartogs, Cantor–Schröder–Bernstein,and König\u003c\/li\u003e \u003c\/ul\u003e  \u003cp\u003eAn excellent textbook for students studying the foundations of mathematics and mathematical proofs, \u003ci\u003eA First Course in Mathematical Logic and Set Theory\u003c\/i\u003e is also appropriate for readers preparing for careers in mathematics education or computer science. In addition, the book is ideal for introductory courses on mathematical logic and\/or set theory and appropriate for upper-undergraduate transition courses with rigorous mathematical reasoning involving algebra, number theory, or analysis.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47988627538149,"sku":"NP9780470905883","price":98.5,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780470905883.jpg?v=1761781030","url":"https:\/\/k12savings.com\/es\/products\/a-first-course-in-mathematical-logic-and-set-theory-isbn-9780470905883","provider":"K12savings","version":"1.0","type":"link"}