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 logicTrue and false statementsLogical connectives and truth tablesLogical equivalenceQuantifiersProofs: Structures and strategiesAxioms, theorems and proofsDirect proofContrapositive proofProof by equivalent statementsProof by casesExistence proofsProof by counterexampleProof by mathematical inductionElementary Theory of Sets. FunctionsAxioms for set theoryInclusion of setsUnion and intersection of setsComplement, difference and symmetric difference of setsOrdered pairs and the Cartersian productFunctionsDefinition and examples of functionsDirect image, inverse imageRestriction and extension of a functionOne-to-one and onto functionsComposition and inverse functions*Family of sets and the axiom of choiceRelationsGeneral relations and operations with relationsEquivalence relations and equivalence classesOrder relations*More on ordered sets and Zorn's lemmaAxiomatic theory of positive integersPeano axioms and additionThe natural order relation and subtractionMultiplication and divisibilityNatural numbersOther forms of inductionElementary number theoryAboslute value and divisibility of integersGreatest common divisor and least common multipleIntegers in base 10 and divisibility testsCardinality. Finite sets, infinite setsEquipotent setsFinite and infinite setsCountable and uncountable setsCounting techniques and combinatoricsCounting principlesPigeonhole principle and parityPermutations and combinationsRecursive sequences and recurrence relationsThe construction of integers and rationals Definition of integers and operationsOrder relation on integersDefinition of rationals, operations and orderDecimal representation of rational numbersThe construction of real and complex numbersThe Dedekind cuts approachThe Cauchy sequences approachDecimal representation of real numbersAlgebraic and transcendental numbersComples numbersThe trigonometric form of a complex number
Valentin Deaconu teaches at University of Nevada, Reno.
Reviews for A Bridge to Higher Mathematics
"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"
