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MATHEMATICAL PROOFS CHARTRAND PDF

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Mathematical Proofs. A Transition to. Advanced Mathematics. Gary Chartrand. Western Michigan University. Albert D. Polimeni. State University of New York at . Mathematical Proofs - 3rd Edition - Chartrand - Ebook download as PDF File .pdf ), Text File .txt) or read book online. textbook. MATHEMATICS. Third Edition. SEYMOUR LIPSCHUTZ, Ph.D. Temple University. MARC LARS LIPSON, Ph.D Sc Third Edition Mathematical Proofs A.


Mathematical Proofs Chartrand Pdf

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Gary Chartrand More on Direct Proof and Proof by Contrapositive Proofs Prove or Disprove Conjectures in Mathematics Revisiting Quantified. Names: Chartrand, Gary, author. | Polimeni Title: Mathematical proofs: a transition to advanced mathematics / Gary Chartrand,. Western .. Presentation slides in PDF and LaTeX formats have been created to accompany. Gary Chartrand is Professor Emeritus of Mathematics at Western Michigan University. He received his Ph.D. in mathematics from Michigan State University.

Even some students who have done well in mathematics say that they are not good with proofs. This, too, is unacceptable. What is required is determination and effort. To have done well on an exam with little or no studying is nothing to be proud of.

Condence based on being well-prepared is good, however. Here is some advice that has worked for several students. First, it is important to understand what goes on in class each day. This means being present and being prepared for every class.

After each class, recopy any lecture notes. When recopying the notes, express sentences in your own words and add details so that everything is as clear as possible.

Mathematical Proofs Solutions Manual

If you run into snags and you will , talk them over with a classmate or your instructor. In fact, its a good idea at least in our opinion to have someone with whom to discuss the material on a regular basis. Not only does it often clarify ideas, it gets you into the habit of using correct terminology and notation.

In addition to learning mathematics from your instructor, solidifying your understanding by careful note-taking and by talking with classmates, your text is or at least should be an excellent source as well.

Read your text carefully with pen or pencil and paper in hand. Make a serious effort to do every homework problem assigned and, eventually, be certain that you know how to solve them.

If there are exercises in the text that have not been assigned, you might even try to solve these as well. Another good idea is to try to create your own problems. In fact, when studying for an exam, try creating your own exam. If you start doing this for all of your classes, you might be surprised at how good you become. Creativity is a major part of mathematics. Discovering mathematics not only contributes to your understanding of the subject but has the potential to contribute to mathematics itself.

Creativity can come in all forms. The following quote is due to the well-known writer J. Rowling author of the Harry Potter novels. Sometimes ideas just come to me. Other times I have to sweat and almost bleed to make ideas come. Its a mysterious process, but I hope I never nd out exactly how it works. In the book Defying Gravity: The Creative Career of Stephen Schwartz from Godspell to Wicked, the author Carol de Giere writes a biography of Stephen Schwartz, one of the most successful composer-lyricists, in which she discusses not only creativity in music but how an idea can lead to something special and interesting and how creative people may have to deal with disappointment.

Indeed, de Giere dedicates her book to the creative spirit within each of us. While he wrote the music for such famous shows as Godspell and Wicked, Schwartz discusses creativity head-on in the song The Spark of Creation he wrote for the musical Children of Eden. In her book, de Giere writes: In many ways, this song expresses the theme of Stephen Schwartzs lifethe naturalness and importance of the creative urge within us.

At the same time he created an anthem for artists. In mathematics our goal is to seek the truth. Finding answers to mathematical questions is important, but we cannot be satised with this alone. We must be certain that we are right and that our explanation for why we believe we are correct is convincing to others.

The reasoning we use as we proceed from what we know to what we wish to show must be logical. It must make sense to others, not just to ourselves. There is joint responsibility here. As writers, it is our responsibility to give an accurate clear argument with enough details provided to allow the reader to understand what we have written and to be convinced. It is the readers responsibility to know the basics of logic and to study the concepts involved so that a well-presented argument will be understood.

Consequently, in mathematics writing is important, very important. Is it really important to write mathematics well?

After all, isnt mathematics mainly equations and symbols? Not at all. It is not only important to write mathematics well, it is important to write well. You will be writing the rest of your life, at least reports, letters and e-mail. Many people who never meet you will know you only by what you write and how you write. Mathematics is a sufciently complicated subject that we dont need vague, hazy and boring writing to add to it. A teacher has a very positive impression of a student who hands in well-written and well-organized assignments and examinations.

You want people to enjoy reading what youve written. It is important to have a good reputation as a writer. Its part of being an educated person. Especially with the large number of e-mail letters that so many of us write, it has become commonplace for writing to be more casual. Although all people would probably subscribe to this since it is more efcient , we should know how to write well formally and professionally when the situation requires it.

You might think that considering how long youve been writing and that youre set in your ways, it will be very difcult to improve your writing. Not really. If you want to improve, you can and will. Even if you are a good writer, your writing can always be improved. Ordinarily, people dont think much about their writing. Often just thinking about your writing is the rst step to writing better.

Chapter 0 What Others Have Said about Writing Many people who are well known in their areas of expertise have expressed their thoughts about writing. Here are quotes by some of these individuals.

Anything that helps communication is good. Anything that hurts it is bad. I like words more than numbers, and I always didconceptual more than computational. Paul Halmos, mathematician Writing is easy. All you have to do is cross out all the wrong words. Mark Twain, author The Adventures of Huckleberry Finn You dont write because you want to say something; you write because youve got something to say.

Scholem Asch, author Either write something worth reading or do something worth writing. Benjamin Franklin, statesman, writer, inventor What is written without effort is in general read without pleasure. Samuel Johnson, writer Easy reading is damned hard writing. Nathaniel Hawthorne, novelist The Scarlet Letter Everything that is written merely to please the author is worthless. The last thing one knows when writing a book is what to put rst. I have made this letter longer because I lack the time to make it short.

The following quote is due to the well-known writer J. Rowling author of the Harry Potter novels. Sometimes ideas just come to me. Other times I have to sweat and almost bleed to make ideas come. Its a mysterious process, but I hope I never nd out exactly how it works. In the book Defying Gravity: The Creative Career of Stephen Schwartz from Godspell to Wicked, the author Carol de Giere writes a biography of Stephen Schwartz, one of the most successful composer-lyricists, in which she discusses not only creativity in music but how an idea can lead to something special and interesting and how creative people may have to deal with disappointment.

Indeed, de Giere dedicates her book to the creative spirit within each of us. While he wrote the music for such famous shows as Godspell and Wicked, Schwartz discusses creativity head-on in the song The Spark of Creation he wrote for the musical Children of Eden. In her book, de Giere writes: In many ways, this song expresses the theme of Stephen Schwartzs lifethe naturalness and importance of the creative urge within us.

At the same time he created an anthem for artists. In mathematics our goal is to seek the truth. Finding answers to mathematical questions is important, but we cannot be satised with this alone. We must be certain that we are right and that our explanation for why we believe we are correct is convincing to others.

The reasoning we use as we proceed from what we know to what we wish to show must be logical. It must make sense to others, not just to ourselves. There is joint responsibility here. As writers, it is our responsibility to give an accurate clear argument with enough details provided to allow the reader to understand what we have written and to be convinced.

It is the readers responsibility to know the basics of logic and to study the concepts involved so that a well-presented argument will be understood.

Consequently, in mathematics writing is important, very important. Is it really important to write mathematics well? After all, isnt mathematics mainly equations and symbols? Not at all. It is not only important to write mathematics well, it is important to write well. You will be writing the rest of your life, at least reports, letters and e-mail. Many people who never meet you will know you only by what you write and how you write.

Mathematics is a sufciently complicated subject that we dont need vague, hazy and boring writing to add to it. A teacher has a very positive impression of a student who hands in well-written and well-organized assignments and examinations. You want people to enjoy reading what youve written.

It is important to have a good reputation as a writer. Its part of being an educated person. Especially with the large number of e-mail letters that so many of us write, it has become commonplace for writing to be more casual.

Although all people would probably subscribe to this since it is more efcient , we should know how to write well formally and professionally when the situation requires it. You might think that considering how long youve been writing and that youre set in your ways, it will be very difcult to improve your writing. Not really. If you want to improve, you can and will. Even if you are a good writer, your writing can always be improved.

Ordinarily, people dont think much about their writing. Often just thinking about your writing is the rst step to writing better. Chapter 0 What Others Have Said about Writing Many people who are well known in their areas of expertise have expressed their thoughts about writing.

Here are quotes by some of these individuals. Anything that helps communication is good. Anything that hurts it is bad. I like words more than numbers, and I always didconceptual more than computational. Paul Halmos, mathematician Writing is easy.

All you have to do is cross out all the wrong words. Mark Twain, author The Adventures of Huckleberry Finn You dont write because you want to say something; you write because youve got something to say. Scholem Asch, author Either write something worth reading or do something worth writing. Benjamin Franklin, statesman, writer, inventor What is written without effort is in general read without pleasure.

Mathematical Proofs - 3rd Edition - Chartrand

Samuel Johnson, writer Easy reading is damned hard writing. Nathaniel Hawthorne, novelist The Scarlet Letter Everything that is written merely to please the author is worthless. The last thing one knows when writing a book is what to put rst. I have made this letter longer because I lack the time to make it short.

Blaise Pascal, mathematician and physicist The best way to become acquainted with a subject is to write a book about it. Benjamin Disraeli, prime minister of England In a very real sense, the writer writes in order to teach himself, to understand himself, to satisfy himself; the publishing of his ideas, though it brings gratication, is a curious anticlimax.

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The skill of writing is to create a context in which other people can think. Edwin Schlossberg, exhibit designer A writer needs three things, experience, observation, and imagination, any two of which, at times any one of which, can supply the lack of the other.

William Faulkner, writer The Sound and the Fury If confusion runs rampant in the passage just read, It may very well be that too much has been said.

Mathematical Writing So thats what he meant!

Then why didnt he say so? Frank Harary, mathematician A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the rst man whom you meet on the street.

David Hilbert, mathematician Everything should be made as simple as possible, but not simpler. Albert Einstein, physicist Never let anything you write be published without having had others critique it. Donald E. Knuth, computer scientist and writer Some books are to be tasted, others to be swallowed, and some few to be chewed and digested.

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Reading maketh a full man, conference a ready man, and writing an exact man. Francis Bacon, writer and philosopher Judge an article not by the quality of what is framed and hanging on the wall, but by the quality of whats in the wastebasket. Anonymous Quote by Leslie Lamport We are all apprentices in a craft where no-one ever becomes a master.

Unfortunately, no one knows what they are. Somerset Maugham, author Of Human Bondage Mathematical Writing Most of the quotes given above pertain to writing in general, not to mathematical writing in particular. However these suggestions for writing apply as well to writing mathematics.

For us, mathematical writing means writing assignments for a mathematics course particularly a course with proofs. Such an assignment might consist of writing a single proof, writing solutions to a number of problems or perhaps writing a term paper which, more than likely, includes denitions, examples, background and proofs.

Well refer to any of these as an assignment. Your goal should be to write correctly, clearly and in an interesting manner. Before you even begin to write, you should have already thought about a number of things.Even some students who have done well in mathematics say that they are not good with proofs. Assume, to the contrary, that a and b are of the same parity. Give an example of three sets A. For a set A. If we had attempted to prove Result 3. We must be certain that we are right and that our explanation for why we believe we are correct is convincing to others.

For P1: Assume that for some positive integer k and any 2k integers a1 , a2 ,.

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