How to Think Like a Mathematician A Companion to Undergraduate Mathematics 1st Edition by Kevin Houston – Ebook PDF Instant Download/Delivery: 978-0521719780, 052171978X
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Product details:
ISBN 10: 052171978X
ISBN 13: 978-0521719780
Author: Kevin Houston
Looking for a head start in your undergraduate degree in mathematics? Maybe you’ve already started your degree and feel bewildered by the subject you previously loved? Don’t panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof – direct method, cases, induction, contradiction and contrapositive – are featured. Concrete examples are used throughout, and you’ll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you’ll soon learn how to think like a mathematician.
Table of contents:
I Study skills for mathematicians
1 Sets and functions
2 Reading mathematics
3 Writing mathematics I
4 Writing mathematics II
5 How to solve problems
II How to think logically
6 Making a statement
7 Implications
8 Finer points concerning implications
9 Converse and equivalence
10 Quantifiers – For all and There exists
11 Complexity and negation of quantifiers
12 Examples and counterexamples
13 Summary of logic
III Definitions, theorems and proofs
14 Definitions, theorems and proofs
15 How to read a definition
16 How to read a theorem
17 Proof
18 How to read a proof
19 A study of Pythagoras’ Theorem
IV Techniques of proof
20 Techniques of proof I: Direct method
21 Some common mistakes
22 Techniques of proof II: Proof by cases
23 Techniques of proof III: Contradiction
24 Techniques of proof IV: Induction
25 More sophisticated induction techniques
26 Techniques of proof V: Contrapositive method
V Mathematics that all good mathematicians need
27 Divisors
28 The Euclidean Algorithm
29 Modular arithmetic
30 Injective, surjective, bijective – and a bit about infinity
31 Equivalence relations
VI Closing remarks
32 Putting it all together
33 Generalization and specialization
34 True understanding
35 The biggest secret
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