Title An Introduction to Mathematical Reasoning; Author(s) Peter J. Eccles pages; eBook PDF files; Language: English; ISBN ; ISBN An introduction to mathematical reasoning: lectures on numbers, sets, and functions / Peter J. Eccles. Includes bibliographical references and index. ISBN 0 Page 1. Page 2. Page 3. Page 4. Page 5. Page 6. Page 7. Page 8. Page 9. Page Page Page Page Page Page Page Page Page
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Categories and Sets - An Introduction to Mathematical Reasoning - by Peter J. Eccles. Peter J. Eccles, University of Manchester PDF; Export citation. Eccles - An Introduction to Mathematical Reasoning - Numbers, Sets and chausifetonis.ml - Ebook download as PDF File .pdf), Text File .txt) or read book online. An Introduction to Mathematical Reasoning - Download as PDF File .pdf), Text File of America by Cambridge University Press. and functions / Peter J. Eccles.
Although some of you may have managed to learn calculus by simply doing the problems, while just skimming back through a minimal amount of the text, cutting corners in this manner simply will not work in this course.
Fortunately, our textbook is very well written, and should amply reward those who take the time to read it properly.
Two different kinds of homework problems will be assigned in this course. Ungraded problems are not to be turned in, but you are expected to solve them carefully, in great detail; failure to so may result in an inability to understand the course material.
Answers to these problems are usually contained in the back of your textbook. Don't peak, though! If you simply look up the answer before seriously trying to find the solution, the only person you will be cheating is yourself. Homework sets, by contrast, will be turned in usually on Tuesdays. Your performance on these assignments will significantly affect your grade in the course. These assignments may be found by clicking on the hyper-link below.
Keep in mind that these assignments are subject to revision until the previous Thursday. You are therefore encouraged to frequently check the relevant web-page for changes.
Refreshing your browser while doing so will ensure that you are really looking at the latest version. Late homework will not be accepted. However, grades for homework assignments may be dropped in cases of documented medical problems or similar diffculties.
By Definition 2. Now we clearly cannot prove this by exhausting all the possibilities one at a time. However, for an integer q, either q 50 or q Hence is not even and so is odd as required. The symbol at the end of this proof is there simply to mark the completion of the proof. I recommend that proofs are concluded by stating what has been proved as occurs here.
This proof might seem over-elaborate, but if you think about it you will see that it is simply spelling out a simple reason why is an odd number. You might like to try to find a better proof based simply on the above definitions.
When the deductive method was first used, for example in the Elements of Euclid written about B. This approach sees mathematics as a body of facts about real objects: points, lines, numbers. These facts are determined by using certain accepted self-evident rules of deduction from certain accepted obvious facts, the axioms. There are a number of philosophical difficulties about this point of view. For another it has been discovered that different sets of axioms are possible and give rise to different theories which appear to be equally valid mathematically.
This is not a mathematical question but is one for physicists and astronomers. However, the awareness that there is more than one possible geometry was enormously liberating and has given rise to much interesting mathematics as well as physics. Without it the geometry which provides the language used in general relativity theory would not have been discovered. Most mathematicians do appear to take the view that mathematics deals with real objects.
We do mathematics by exploring what follows from the truth of these axioms using certain accepted rules of deduction. The reader will meet good examples of this axiomatic method in the study of algebra. Since this book is primarily concerned with introducing the reader to mathematical reasoning and exposition it seems best to keep the mathematical context very familiar and so we will mainly restrict ourselves to arithmetic.
It would be cumbersome to develop these properties formally from a list of axioms although this can be done. We will simply assume that the reader is familiar with the basic algebraic properties of numbers dealing with addition, subtraction, multiplication and division and take these for granted. These can be summarized as follows. Properties 2. The operations of sum or addition and product or multiplication of real numbers have the following properties.
This tells us how to remove brackets.
All the usual algebraic properties of numbers can be deduced from the above statements. The distinction between rational numbers and real numbers will be considered in Chapter Most of this book will be concerned with the integers. We will also make use of the order properties of the real numbers and have already made use of these in the proof of Proposition 2.
Exercises 2. This is of course a valid argument but assumes the properties of the rational numbers. For the present purpose of illustrating the use of definitions it is appropriate to seek an argument based simply on the integers. For details of such an axiomatic approach see for example G. Birkhoff and S. Mac Lane A survey of modern algebra, Macmillan, Fourth edition The steps of the logical argument are provided by implications.
One of the main aims of this book is to describe a variety of methods of proof so that you can follow these when you meet them and also construct proofs for yourself.
No doubt anyone reading this book will have been seeing and understanding proofs for years. At university you are expected to be able to construct your own proofs and, as importantly, to write them out carefully so that other people can understand them — or even so that you can understand them yourself when you come to look back at your.
One real difficulty is that we do not normally discover proofs in the polished form in which they are presented.
It is important to realize that you will usually spend time constructing a proof before you then write out a formal proof. You can think of this as erecting a sort of scaffolding for the purpose of constructing the proof.
When the proof has been constructed the scaffolding is removed so that the proof can be admired in all its economical beautiful simplicity! However, one difficulty for the person encountering the proof for the first time is that it can be hard to make sense of.
To read and understand the proof we may have to reconstruct the scaffolding for ourselves from the formal proof. This can be difficult — but not usually as difficult as thinking of the proof in the first place unless the proof is very badly written. This is a problem not just for beginning undergraduates but also for professional mathematicians when they read mathematics.
You may ask why then the scaffolding is not retained. The difficulty is that if every detail is given then mathematical arguments become enormously long and cluttered. The aim is to pitch your writing at the level of the expected reader so that there is just enough information to enable the reconstruction of the scaffolding if necessary but not so much that it would mask the essence of the argument. Too much detail can make a proof based possibly on one simple idea appear enormously complicated.
It is, however, important not to use such phrases as a lazy way of avoiding thinking about the details. It is also the case that excessive pedantic precision can sometimes make mathematics hard to read. Writing mathematics is not like writing a computer program; what is written will be read by a human being who has much common experience with the writer and so is able to anticipate to some extent what the writer intends.
As Gila Hanna has written, The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight. While learning to write good mathematics it is probably better to err on the side of pedantry. When you read most mathematics books you need to work with pencil in hand reconstructing the detail behind the proofs provided.
You cannot normally read a mathematics book like a conventional novel. Very many theorems are of the form P Q. How do we set about proving such a statement? Since the statement is necessarily true if P is false remember Table 2. Then from the truth table we see that P Q is true so long as Q is also true. So to prove that P Q is true, it is sufficient to assume that P is true and deduce Q. This is the direct form of proof.
Here is an example. Proposition 3. Constructing a proof. We can summarize what is needed in the following way using a given—goal diagram. This leads to the following new given-goal diagram. We now start to think how we can obtain something like the goal from the given statements.
We see that we want a2 and b2 in the goal and this suggests multiplying the given inequality through by a and by b. Thus and using the fact that a and b are both positive.
The reader might well ask where the three numbered implications have come from; why are these true? The answer is that these statements follow immediately from fundamental properties of, the real numbers which can be encapsulated in the inequality or order axioms. We have already commented that it would be unwieldy to work from a complete axiom system in this book and readers are unlikely to have difficulties with the algebraic properties assumed.
However, inequalities are less familiar and so it seems useful to list the basic properties to be assumed. Axioms 3. For real numbers a, band c, iii Multiplication law. For real numbers a, band c, iv Transitive law. For real numbers a, band c, Now statements 3. We can now write out a formal proof of Proposition 3.
We do sometimes use more symbols so that the above proof might have been written out as follows. This presentation highlights the fact that in constructing a proof as a chain of implications we repeatedly use which is readily checked by using a truth table. One problem in writing out proofs is to decide how much detail to give and what can be assumed. There is no simple answer to this.
Although the above proof did start from the inequality axioms this was not explicitly referred to in the formal proof. You always do have to start somewhere. But it is cumbersome to reduce everything to a set of axioms and there is usually a wide body of results which it is reasonable to assume. For example the result of Proposition 2.
It was explored in Chapter 2 simply to illustrate the role of definitions. It is good style to indicate where the hypotheses in the result being proved are used in the proof.
Also if some steps in the argument are only valid under certain conditions then you should verify that these conditions are indeed satisfied. Proof by cases When proving universal implications it is often very difficult to consider together all the objects satisfying the hypothesis.
Here is a very simple example. Example 3.
This is not always possible. Recall the proof of Proposition 2. It is impossible to work through the integers one by one.