218 Pages 37 B/W Illustrations
    by Chapman & Hall

    220 Pages
    by Chapman & Hall

    218 Pages 37 B/W Illustrations
    by Chapman & Hall

    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.

    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

     

    Biography

    Valentin Deaconu teaches at University of Nevada, Reno.

    This is one of the shorter books for a course that introduces students to the concept of mathematical proofs. The brevity is due to the "bare-bones" nature of the treatment. The number of topics covered, the number of examples, and the number of exercises are not smaller than what appears in competing textbooks; what is shorter is the text that one finds between theorems, lemmas, examples, and exercises. Besides the topics found in similar textbooks (i.e., proof techniques, logic, set theory, relations, and functions), there are chapters on (very) elementary number theory, combinatorial counting techniques, and Peano axioms on the set of positive integers. Several chapters are devoted to the construction of various kinds of numbers, such as integers, rationals, real numbers, and complex numbers. Answers to around half the exercises are included at the end of the book, and a few have complete solutions. This reviewer finds the book more enjoyable than the average competing textbook.

    --M. Bona, University of Florida