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
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
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