How to Think Like a Mathematician: A Companion to Undergraduate MathematicsLooking 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. |
Contents
Sets and functions | 3 |
Reading mathematics | 14 |
Writing mathematics I | 21 |
How to solve problems | 41 |
How to think logically | 51 |
Making a statement | 53 |
Implications | 63 |
Finer points concerning implications | 69 |
How to read a proof | 119 |
A study of Pythagoras Theorem | 126 |
Techniques of proof | 137 |
Direct method | 139 |
Some common mistakes | 149 |
Proof by cases | 155 |
Contradiction | 161 |
Induction | 166 |
Converse and equivalence | 75 |
Quantifiers For all and There exists | 80 |
Complexity and negation of quantifiers | 84 |
Examples and counterexamples | 90 |
Summary of logic | 96 |
Definitions theorems and proofs | 97 |
Definitions theorems and proofs | 99 |
How to read a definition | 103 |
How to read a theorem | 109 |
Proof | 116 |
Divisors | 187 |
The Euclidean Algorithm | 196 |
Modular arithmetic | 208 |
Injective surjective bijective and a bit about infinity | 218 |
Equivalence relations | 230 |
Putting it all together | 243 |
True understanding | 252 |
B Commonly used symbols and notation | 258 |
Other editions - View all
How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston Limited preview - 2009 |
How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston No preview available - 2009 |
How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston No preview available - 2009 |
Common terms and phrases
answer apply assume bijective bijective functions calculate Chapter consider contrapositive converse Cosine Rule countable counterexample define definition denoted digit divides Division Lemma elements empty set equal equation equivalence classes equivalence relation Euclid's Lemma example Exercises Exercises exists Fermat's Fermat's Little Theorem finite set formula function f give given by f greatest common divisor Hence idea implies inductive step infinite number injective inverse irrational look Mathematical Induction mathematician means method modular arithmetic natural numbers negation non-examples not(A not(B Note obvious odd numbers positive integer prime numbers produce proof by contradiction prove Pythagoras quantifiers rational numbers real numbers rewrite right-angled triangle sentence square root statement is false statement is true subset Summary Suppose surjective symbols techniques theorem thinking trivial true statement truth table understand Winston Churchill words write zero