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.
Table of Contents
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.
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.
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.