A TRANSITION TO ADVANCED MATHEMATICS helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful text of its kind, the 7th edition continues to provide a firm foundation in major concepts needed for continued study and guides students to think and express themselves mathematically—to analyze a situation, extract pertinent facts, and draw appropriate conclusions. The authors place continuous emphasis throughout on improving students' ability to read and write proofs, and on developing their critical awareness for spotting common errors in proofs. Concepts are clearly explained and supported with detailed examples, while abundant and diverse exercises provide thorough practice on both routine and more challenging problems. Students will come away with a solid intuition for the types of mathematical reasoning they'll need to apply in later courses and a better understanding of how mathematicians of all kinds approach and solve problems.
This text was developed from lecture notes for a course at Central Michigan U. that was designed to bridge the gap between calculus and advanced courses for students who say: "I understand mathematics, but I just can't do proofs." Provides an overview of the major ideas needed for continued work, guides students to think and express themselves mathematically, and presents an introduction to modern algebra and analysis, including the foundational topics of logic, sets, relations, and functions. To help make the introduction to elementary proof techniques more manageable, new to this edition are separate sections on direct proofs and proofs by contrapositive and contradiction. There are also new and revised explanations, examples, and exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Product dimensions: 7.50 (w) x 9.30 (h) x 0.90 (d)
Meet the Author
The authors are the leaders in this course area. They decided to write this text based upon a successful transition course that Richard St. Andre developed at Central Michigan University in the early 1980s. This was the first text on the market for a transition to advanced mathematics course and it has remained at the top as the leading text in the market. Douglas Smith is Professor of Mathematics at the University of North Carolina at Wilmington. Dr. Smith's fields of interest include Combinatorics / Design Theory (Team Tournaments, Latin Squares, and applications), Mathematical Logic, Set Theory, and Collegiate Mathematics Education.
Maurice Eggen is Professor of Computer Science at Trinity University. Dr. Eggen's research areas include Parallel and Distributed Processing, Numerical Methods, Algorithm Design, and Functional Programming.
Richard St. Andre is Associate Dean of the College of Science and Technology at Central Michigan University. Dr. St. Andre's teaching interests are quite diverse with a particular interest in lower division service courses in both mathematics and computer science.
1. LOGIC AND PROOFS. Propositions and Connectives. Conditionals and Biconditionals. Quantifiers. Basic Proof Methods I. Basic Proof Methods II. Proofs Involving Quantifiers. Additional Examples of Proofs 2. SET THEORY. Basic Notions of Set Theory. Set Operations. Extended Set Operations and Indexed Families of Sets. Induction. Equivalent Forms of Induction. Principles of Counting. 3. RELATIONS AND PARTITIONS. Relations. Equivalence Relations. Partitions. Ordering Relations. Graphs. 4. FUNCTIONS. Functions as Relations. Constructions of Functions. Functions That Are Onto; One-to-One Functions. One-to-One Correspondences and Inverse Functions. Images of Sets. Sequences. 5. CARDINALITY. Equivalent Sets; Finite Sets. Infinite Sets. Countable Sets. The Ordering of Cardinal Numbers. Comparability of Cardinal Numbers and the Axiom of Choice. 6. CONCEPTS OF ALGEBRA: GROUPS. Algebraic Structures. Groups. Subgroups. Operation Preserving Maps. Rings and Fields. 7. CONCEPTS OF ANALYSIS: COMPLETENESS OF THE REAL NUMBERS. Ordered Field Properties of the Real Numbers. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. The Bounded Monotone Sequence Theorem. Comparability of Cardinals and the Axiom of Choice.