# A Transition to Abstract Mathematics: Learning Mathematical Thinking and Writing

Language: English

Pages: 384

ISBN: B005FP4ZHE

Format: PDF / Kindle (mobi) / ePub

Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. Mathematical Thinking and Writing teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.

Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure.

After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.

* Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction

* Explains identification of techniques and how they are applied in the specific problem

* Illustrates how to read written proofs with many step by step examples

* Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter

* The Instructors Guide and Solutions Manual points out which exercises simply must be either assigned or at least discussed because they undergird later results

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principles of algebraic manipulation follow from these assumptions and to learn to write proofs of such theorems. 2.1 Basic Algebraic Properties of Real Numbers We begin by writing proofs of some of the most basic algebraic properties of real numbers. We will remind ourselves of the relevant assumptions as we need them. The properties of equality are of primary importance. (A1) Properties of equality: (a) For all a ∈ R, a = a. (b) For all a, b ∈ R, if a = b, then b = a. (c) For all a, b, c ∈ R,

p/q. 106 Chapter 3 3.9 Sets and Their Properties Roots of Real Numbers Up to this point, we have made no reference to assumptions A20–A22 from Chapter 0. In this section, we want to explore A22: (A22) For every positive real number x and any positive integer n, there exists a real solution y to the equation yn = x. Such a solution y is called an nth root of x. You might wonder why assumption A22 refers to the solution of the equain order to address roots of real numbers instead of using the

irrational numbers at ﬁrst, and now is a good time to touch brieﬂy on some of their views. 3.10 Irrational Numbers 109 To the Greeks, numbers were conceived in terms of the geometric concepts of length, area, and volume, all of which were constructible according to certain idealized rules. They had very sophisticated techniques for imagining their idealized construction using the only two geometric shapes they considered perfect: straight line segments (crudely constructible with a

necessary for each path, and which may be safely omitted without leaving any holes in the logical progression. Of course, even if a particular result is necessary later, one might decide that to omit its proof details does not deprive the students of a valuable learning experience. The instructor might choose simply to elaborate on how one would go about proving a certain theorem, then allow the students to use it as if they had proved it themselves. 2. Cover Part I in its entirety, saving

Here are several pairs of sets. (a) {1, 2, 3, 4} and {a, b, c} (b) {1, 2, 3, 4} and N (c) Z and Z (d) Z and W (e) R and R For each of the above pairs of sets, ﬁnd four functions {f1 , f2 , f3 , f4 } from the ﬁrst set to the second set with the following properties, if such functions are possible. It is not necessary to prove that your functions have the desired properties, but be prepared to provide as much explanation as necessary to support your claim. (i) f1 is one-to-one but not onto. (ii)