A bridge to higher mathematics

著者

    • Deaconu, Valentin
    • Pfaff, Donald C. (Donald Chesley)

書誌事項

A bridge to higher mathematics

Valentin Deaconu, Donald C. Pfaff

（Textbooks in mathematics）

CRC Press, c2017

  • : paperback

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注記

Includes bibliographical references and index

内容説明・目次

内容説明

A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof. For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn's lemma and the axiom of choice are included. More challenging problems are marked with a star. All these materials are optional, depending on the instructor and the goals of the course.

目次

Elements of logic True and false statements Logical connectives and truth tables Logical equivalence Quantifiers Proofs: Structures and strategies Axioms, theorems and proofs Direct proof Contrapositive proof Proof by equivalent statements Proof by cases Existence proofs Proof by counterexample Proof by mathematical induction Elementary Theory of Sets. Functions Axioms for set theory Inclusion of sets Union and intersection of sets Complement, difference and symmetric difference of sets Ordered pairs and the Cartersian product Functions Definition and examples of functions Direct image, inverse image Restriction and extension of a function One-to-one and onto functions Composition and inverse functions *Family of sets and the axiom of choice Relations General relations and operations with relations Equivalence relations and equivalence classes Order relations *More on ordered sets and Zorn's lemma Axiomatic theory of positive integers Peano axioms and addition The natural order relation and subtraction Multiplication and divisibility Natural numbers Other forms of induction Elementary number theory Aboslute value and divisibility of integers Greatest common divisor and least common multiple Integers in base 10 and divisibility tests Cardinality. Finite sets, infinite sets Equipotent sets Finite and infinite sets Countable and uncountable sets Counting techniques and combinatorics Counting principles Pigeonhole principle and parity Permutations and combinations Recursive sequences and recurrence relations The construction of integers and rationals Definition of integers and operations Order relation on integers Definition of rationals, operations and order Decimal representation of rational numbers The construction of real and complex numbers The Dedekind cuts approach The Cauchy sequences approach Decimal representation of real numbers Algebraic and transcendental numbers Comples numbers The trigonometric form of a complex number

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