An Accompaniment to Higher MathematicsFor Students Congratulations! You are about to take a course in mathematical proof. If you are nervous about the whole thing, this book is for you (if not, please read the second and third paragraphs in the introduction for professors following this, so you won't feel left out). The rumors are true; a first course in proof may be very hard because you will have to do three things that are probably new to you: 1. Read mathematics independently. 2. Understand proofs on your own. :1. Discover and write your own proofs. This book is all about what to do if this list is threatening because you "never read your calculus book" or "can't do proofs. " Here's the good news: you must be good at mathematics or you wouldn't have gotten this far. Here's the bad news: what worked before may not work this time. Success may lie in improving or discarding many habits that were good enough once but aren't now. Let's see how we've gotten to a point at which someone could dare to imply that you have bad habits. l The typical elementary and high school mathematics education in the United States tends to teach students to have ineffective learning habits, 1 In the first paragraph, yet. xiv Introduction and we blush to admit college can be just as bad. |
Contents
Examples | 1 |
Informal Language and Proof | 2 |
27 | 7 |
2 | 37 |
42 | 81 |
Formal Language and Proof | 106 |
Laboratories | 117 |
A Theoretical Apologia | 147 |
73 | 158 |
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Common terms and phrases
Algebra Aon Bo arbitrary sets argument assume bijection calculus called Chapter codomain conclusion condition construct continuous contraposition course deduce defined definition of injective denoted direct proof discussion domain domain(f equivalence relation Exercise exists Explore family of sets formula function f ƒ and g give Given sets go f graph Here's hint available hypothesis identity element implies indexed family induction injective function intersection inverse language least logic look mathematicians mathematics Mean Value Theorem non-examples nonempty notation Note ordered pairs pair in f picture problem proof by contradiction proof by contraposition proof form proof structure prove reader reading real numbers real-valued functions Recall satisfying Section sequence set of ordered sort square function statement form subset Suppose ƒ sure surjective surjective function things triangle true truth table union universal quantifiers variable vertex vertices Vx(x write