A Transition to Advanced Mathematics

by Smith, Douglas; Eggen, Maurice; St. Andre, Richard

• ISBN13: 9780495562023
• ISBN10: 0495562025
• Edition: 7th
• Format: Hardcover
• Copyright: 2010-06-01
• Publisher: Brooks Cole

Summary

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

Quantifiers

Basic Proof Methods I

Basic Proof Methods II

Proofs Involving Quantifiers

Additional Examples of Proofs

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

Relations And Partitions

Relations

Equivalence Relations

Partitions

Ordering Relations

Graphs

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

Cardinality

Equivalent Sets

Finite Sets

Infinite Sets

Countable Sets

The Ordering of Cardinal Numbers

Comparability of Cardinal Numbers and the Axiom of Choice

Concepts Of Algebra: Groups

Algebraic Structures

Groups

Subgroups

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

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