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This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction) then applies them to easily-understood questions in elementary number theory and counting then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics.
Topics are addressed in the context of familiar objects easily-understood, engaging examples and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity.
Preface for the Instructor.
Preface for the Student.
Elementary concepts.
Numbers, Sets and Functions.
Language and Proofs.
Induction.
Bijections and Cardinality.
Properties of numbers.
Combinatorial Reasoning.
Divisibility.
Modular Arithmetic.
The Rational Numbers.
Discrete mathematics.
Probability.
Two Principles of Counting.
Graph Theory.
Recurrence Relations.
Continuous mathematics.
The Real Numbers.
Sequences and Series.
Continuous Functions.
Differentiation.
Integration.
The Complex Numbers.
From N to R.
Hints for Selected Exercises.
Suggestions for Further Reading.
List of Notation.
Index.
For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics