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## Details

Genre/Form: | Problems and exercises Textbooks |
---|---|

Document Type: | Book |

All Authors / Contributors: |
Gary Chartrand; Albert D Polimeni; Ping Zhang |

ISBN: | 9780321797094 0321797094 9780321782519 0321782518 |

OCLC Number: | 793099426 |

Description: | xiii, 400 pages : illustrations ; 24 cm |

Contents: | 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 -- Sets : Describe a set ; Subsets ; Set operations ; Indexed collections of sets ; Partitions of sets ; Cartesian products of sets -- Logic : Statements ; The negation of a statement ; The disjunction and conjunction of statements ; The implication ; More on implications ; The biconditional ; Tautologies and contradictions ; Logical equivalence ; Some fundamental properties of logical equivalence ; Quantified statements ; Characterizations of statements -- Direct proof and proof by contrapositive : Trivial and vacuous proofs ; Direct proofs ; Proof by contrapositive ; Proof by cases ; Proof evaluations -- More on direct proof and proof by contrapositive : Proofs involving divisibility of integers ; Proofs involving congruence of integers ; Proofs involving real numbers ; Proofs involving sets ; Fundamental properties of set operations ; Proofs involving Cartesian products of sets -- Existence and proof by contradiction : Counterexamples ; Proof by contradiction ; A review of three proof techniques ; Existence proofs ; Disproving existence statements -- Mathematical induction : The principle of mathematical induction ; A more general principle of mathematical induction ; The strong principle of mathematical induction -- Prove or disprove : Conjectures in mathematics ; Revisiting quantified statements ; Testing statements -- Equivalence relations : Relations ; Properties of relations ; Equivalence relations ; Properties of equivalence classes ; Congruence modulo n ; The integers modulo n -- Functions : The definition of function ; The set of all functions from A to B ; One-to-one and onto functions ; Bijective functions ; Composition of functions ; Inverse functions ; Permutations -- Cardinalities of sets : Numerically equivalent sets ; Denumerable sets ; Uncountable sets ; Comparing cardinalities of sets ; The Schroder-Bernstein theorem -- Proofs in number theory : Divisibility properties of integers ; The division algorithm ; Greatest common divisors ; The Euclidean algorithm ; Relatively prime integers ; The fundamental theorem of arithmetic ; Concepts involving sums of divisors -- Proofs in calculus : Limits of sequences ; Infinite series ; Limits of functions ; Fundamental properties of limits of functions ; Continuity ; Differentiability -- Proofs in group theory : Binary operations ; Groups ; Permutations groups ; Fundamental properties of groups ; Subgroups ; Isomorphic groups -- Proofs in ring theory online : Rings ; Elementary properties of rings ; Subrings ; Integral domains ; Fields -- Proofs in linear algebra online : Properties of vectors in 3-space : Vector spaces ; Matrices ; Some properties of vector spaces ; Subspaces ; Spans of vectors ; Linear dependence and independence ; Linear transformations ; Properties of linear transformations -- Proofs in topology online : Metric spaces ; Open sets in metric spaces ; Continuity in metric spaces ; Topological spaces ; Continuity in topological spaces. |

Responsibility: | Gary Chartrand, Albert D. Polimeni, Ping Zhang. |

More information: |

### Abstract:

This book prepares students for the more abstract mathematics courses that follow calculus. The author introduces students to proof techniques, analyzing proofs, and writing proofs of their own. It also provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory.

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