Mathematical Proofs – Chantrad, Polimeni, Zhang – 2nd Edition


Mathematical Proofs: A Transition to Advanced Mathematics, Second Edition, prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own.

As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.

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  • 0. Communicating Mathematics
    Learning Mathematics
    What Others Have Said About Writing
    Mathematical Writing
    Using Symbols
    Writing Mathematical Expressions
    Common Words and Phrases in Mathematics
    Some Closing Comments about Writing
    1. Sets
    1.1 Describing a Set
    1.2 Subsets
    1.3 Set Operations
    1.4 Indexed Collections of Sets
    1.5 Partitions of Sets
    1.6 Cartesian Products of Sets
    Exercises for Chapter 1
    2. Logic
    2.1 Statements
    2.2 The Negation of a Statement
    2.3 The Disjunction and Conjunction of Statements
    2.4 The Implication
    2.5 More on Implications
    2.6 The Biconditional
    2.7 Tautologies and Contradictions
    2.8 Logical Equivalence
    2.9 Some Fundamental Properties of Logical Equivalence
    2.10 Quantified Statements
    2.11 Characterizations of Statements
    Exercises for Chapter 2
    3. Direct Proof and Proof by Contrapositive
    3.1 Trivial and Vacuous Proofs
    3.2 Direct Proofs
    3.3 Proof by Contrapositive
    3.4 Proof by Cases
    3.5 Proof Evaluations
    Exercises for Chapter 3
    4. More on Direct Proof and Proof by Contrapositive
    4.1 Proofs Involving Divisibility of Integers
    4.2 Proofs Involving Congruence of Integers
    4.3 Proofs Involving Real Numbers
    4.4 Proofs Involving Sets
    4.5 Fundamental Properties of Set Operations
    4.6 Proofs Involving Cartesian Products of Sets
    Exercises for Chapter 4
    5. Existence and Proof by Contradiction
    5.1 Counterexamples
    5.2 Proof by Contradiction
    5.3 A Review of Three Proof Techniques
    5.4 Existence Proofs
    5.5 Disproving Existence Statements
    Exercises for Chapter 5
    6. Mathematical Induction
    6.1 The Principle of Mathematical Induction
    6.2 A More General Principle of Mathematical Induction
    6.3 Proof by Minimum Counterexample
    6.4 The Strong Principle of Mathematical Induction
    Exercises for Chapter 6
    7. Prove or Disprove
    7.1 Conjectures in Mathematics
    7.2 Revisiting Quantified Statements
    7.3 Testing Statements
    7.4 A Quiz of "Prove or Disprove" Problems
    Exercises for Chapter 7
    8. Equivalence Relations
    8.1 Relations
    8.2 Properties of Relations
    8.3 Equivalence Relations
    8.4 Properties of Equivalence Classes
    8.5 Congruence Modulo n
    8.6 The Integers Modulo n
    Exercises for Chapter 8
    9. Functions
    9.1 The Definition of Function
    9.2 The Set of All Functions from A to B
    9.3 One-to-one and Onto Functions
    9.4 Bijective Functions
    9.5 Composition of Functions
    9.6 Inverse Functions
    9.7 Permutations
    Exercises for Chapter 9
    10. Cardinalities of Sets
    10.1 Numerically Equivalent Sets
    10.2 Denumerable Sets
    10.3 Uncountable Sets
    10.4 Comparing Cardinalities of Sets
    10.5 The Schröder-Bernstein Theorem
    Exercises for Chapter 10
    11. Proofs in Number Theory
    11.1 Divisibility Properties of Integers
    11.2 The Division Algorithm
    11.3 Greatest Common Divisors
    11.4 The Euclidean Algorithm
    11.5 Relatively Prime Integers
    11.6 The Fundamental Theorem of Arithmetic
    11.7 Concepts Involving Sums of Divisors
    Exercises for Chapter 11
    12. Proofs in Calculus
    12.1 Limits of Sequences
    12.2 Infinite Series
    12.3 Limits of Functions
    12.4 Fundamental Properties of Limits of Functions
    12.5 Continuity
    12.6 Differentiability
    Exercises for Chapter 12
    13. Proofs in Group Theory
    13.1 Binary Operations
    13.2 Groups
    13.3 Permutation Groups
    13.4 Fundamental Properties of Groups
    13.5 Subgroups
    13.6 Isomorphic Groups
    Exercises for Chapter 13
    Answers and Hints to Selected Odd-Numbered Exercises
    Index of Symbols
    Index of Mathematical Terms
  • Citation
    • Full Title: Mathematical Proofs
    • Author/s:
    • ISBN-13: 9780321390530
    • Edition: 2nd Edition
    • Topic: Math
    • Subtopic: Mathematics
    • File Type: Solution Manual
    • Idioma: English

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