Table of Contents

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

About the Author

Valentin Deaconu teaches at University of Nevada, Reno.

Reviews

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