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A TRANSITION TO ADVANCED MATHEMATICS helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful book of its kind, the 7th edition continues to provide a firm foundation in major concepts needed for continued study and guides users 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 users' 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
Logic And Proofs
Propositions and Connectives
Conditionals and Biconditionals
Basic Proof Methods I
Basic Proof Methods II
Proofs Involving Quantifiers
Additional Examples of Proofs
Basic Notions of Set Theory
Extended Set Operations and Indexed Families of Sets
Equivalent Forms of Induction
Principles of Counting
Relations And Partitions
Functions as Relations
Constructions of Functions
Functions That Are Onto
One-to-One Correspondences and Inverse Functions
Images of Sets.
The Ordering of Cardinal Numbers
Comparability of Cardinal Numbers and the Axiom of Choice
Concepts Of Algebra: Groups
Operation Preserving Maps
Rings and Fields
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
Table of Contents provided by Publisher. All Rights Reserved.