A Transition to Advanced Mathematics by Smith

• Foundations
• micromass
In summary: This book is to help them with the transition, and I am not aware of any other books that deliberately address this problem.In summary, "A Transition to Advanced Mathematics" by Douglas Smith, Maurice Eggen, and Richard St. Andre is a textbook designed for undergraduates to help them transition from calculus to more abstract mathematics. It covers topics such as logic, proofs, set theory, relations, functions, cardinality, and algebra and analysis concepts. There are also sections on graph theory and worked out examples. While there are other books that cover similar material, this book specifically focuses on the transition to advanced mathematics and is recommended for

For those who have used this book

• Lightly don't Recommend

• Total voters
4
micromass
Staff Emeritus
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• Preface
• Preface to the Student
• Logic and Proofs
• Propositions and Connectives
• Conditionals and Biconditionals
• Quantifiers
• Basic Proof Methods I
• Basic Proof Methods II
• Proofs Involving Quantifiers
• Set Theory
• Basic Concepts of Set Theory
• Set Operations
• Extended Set Operations and Indexed Families of Sets
• Mathematical Induction
• Equivalent Forms of Induction
• Principles of Counting
• Relations and Partitions
• Cartesian Products and Relations
• Equivalence Relations
• Partitions
• Ordering Relations
• Graphs
• Functions
• Functions as Relations
• Constructions of Functions
• Functions That Are Onto; One-to-One Functions
• One-to-One Correspondences and Inverse Functions
• Images of Sets
• Sequences
• Cardinality
• Equivalent Sets; Finite Sets
• Infinite Sets
• Countable Sets
• The Ordering of Cardinal Numbers
• Comparability of Cardinal Numbers and the Axiom of Choice
• Concepts of Algebra
• Algebraic Structures
• Groups
• Subgroups
• Operation Preserving Maps
• Rings and Fields
• Concepts of Analysis
• Completeness of the Real Numbers
• The Heine–Borel Theorem
• The Bolzano–Weierstrass Theorem
• The Bounded Monotone Sequence Theorem
• Equivalents of Completeness
• Index

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micromass - correct me if I'm wrong, but I think I would put this under "foundations," as it's used in a proofs course which usually comes after the calculus sequence but before proofs based classes like topology. So I'm not sure if it qualifies as "intro math." Topics include set theory, logic, etc.

dkotschessaa said:
micromass - correct me if I'm wrong, but I think I would put this under "foundations," as it's used in a proofs course which usually comes after the calculus sequence but before proofs based classes like topology. So I'm not sure if it qualifies as "intro math." Topics include set theory, logic, etc.

Thanks, I changed it!

Having said that, this is a great book for such a class, IMO. I realize not all universities have such a course and that one is often thrown into proof writing in a higher level course, in which case this book would make a good companion, I would think.

This book has sections on Concepts of Algebra and Concepts of Analysis,my book How to prove it by velleman doesn't have that ,does that mean this book is better?

theoristo said:
This book has sections on Concepts of Algebra and Concepts of Analysis,my book How to prove it by velleman doesn't have that ,does that mean this book is better?

Also graph theory!

But no, this book (Smith's) is a textbook, written for classroom use. The author's assumption is that somebody will be teaching you the material. It is rather dense, so it contains more topics, but is still smaller than Velleman's book (in actual pages and size).

Velleman's book is more of a self study guide. It contains a great deal more of written explanation, and a lot more worked out examples.

-Dave K

dkotschessaa said:
Also graph theory!

But no, this book (Smith's) is a textbook, written for classroom use. The author's assumption is that somebody will be teaching you the material. It is rather dense, so it contains more topics, but is still smaller than Velleman's book (in actual pages and size).

Velleman's book is more of a self study guide. It contains a great deal more of written explanation, and a lot more worked out examples.

-Dave K

what about this? https://www.amazon.com/dp/0321390539/?tag=pfamazon01-20 main purpose is to learn proof in calculus and analysis ,which one is the most recommended.

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theoristo said:
what about this? https://www.amazon.com/dp/0321390539/?tag=pfamazon01-20 main purpose is to learn proof in calculus and analysis ,which one is the most recommended.

I don't know anything about it myself. I see you have asked for a thread to be created for it, but it's in with a list of other books, so I don't know how soon the mods will be able to create all those.

-Dave K

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Mathematical Proofs: A Transition to Advanced Mathematics contains sections named Proof in calculus,Proof in group theory,Proof in number theory,is that helpful for analysis?

theoristo said:
Mathematical Proofs: A Transition to Advanced Mathematics contains sections named Proof in calculus,Proof in group theory,Proof in number theory,is that helpful for analysis?

Any experience doing proofs should help with analysis, one hopes. But I haven't completed analysis yet, so hope someone else can answer better.

1 person
I don't really understand the purpose of this book. It also is not very good. I understand that sadly some students even after three to five years long mathematics courses have not been introduced to basic notions and notations. This problem could be rectified in the context of useful and interesting mathematics instead of dull pointless symbol pushing. Simply many people will not benefit much from reading this, the few who will would benefit more from a better book.

lurflurf said:
I don't really understand the purpose of this book. It also is not very good. I understand that sadly some students even after three to five years long mathematics courses have not been introduced to basic notions and notations. This problem could be rectified in the context of useful and interesting mathematics instead of dull pointless symbol pushing. Simply many people will not benefit much from reading this, the few who will would benefit more from a better book.

There is a process of weeding out that takes place in higher mathematics. Either one learns the logic and notations and so forth as one goes along, and is considered to have potential as a mathematician, or one becomes immediately baffled by the transition to more abstract mathematics, from earlier calculation based courses, and drops out, changes majors, or has a much greater struggle getting through.

The ones not weeded out by this process tend to question the need for such books. They assume, perhaps, that students that can't pick it up as they go along aren't suited for higher maths in the first place, so it's just as well.

Myself, having been away from mathematics for over a decade, braced for the transition with Velleman's book, and really enjoyed taking the class using Smith's. I enjoyed the entire process immensely. I loved "pushing symbols" or the exercise of logic on paper. It's not devoid of actual mathematics, but uses examples from a variety of topics, set theory of course, number theory, etc.

I don't know which of the above category I'd be in if I hadn't done this. However, I certainly found the transition less shocking, and I'm doing better in higher math then I did in calculation based courses.

-Dave K

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^This weeding you describe is an artifact of a broken curriculum. A person with significant exposure to mathematics should have some exposure to what the book covers. The fact that it is possible for a person to take five or more yearlong college courses in mathematics without knowing these things is unfortunate, these things should be sprinkled throughout. I often judge a book on a fraction of its contents, this one I have had the misfortune of reading in it entirety. I understand completely that a person can find themselves with some gaps that need filling. The book is most useful (or least useless) to those most unfamiliar with its contents. I would still not recommend it to those people. It is also commonly read by those who know well the contents but want a review or expansion of their knowledge or those with exposure to the content who are having difficulties. The book is even less effective for these individuals. This book many be helpful to many people (though I have my doubts) but I find it hard to believe there are many people for which it is well suited.

$$\aleph \in \ni \neg \sim \mathbb{NZRC} \Rightarrow \Leftrightarrow \wedge \vee \bot \forall \exists \equiv$$
The book is neither necessary or sufficient for that as many symbols are not included and many books that employ such symbols have a summary of notation that describes their meaning. The Smith book while not entirely worthless is inefficient. There is the price 150$for 1.50$ worth of material. I think a book of this length could could cover what it does and a lot more. Defenders claim the book would then be too dense. I think a book more grounded in useful and interesting mathematics would be more interesting and easier to understand because its ideas would be placed in context. As is often the case in mathematics learning two related things (in this case useful and interesting mathematics and symbols, logic, and proofs) simultaneously is easier and more productive.

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1. What is the main purpose of "A Transition to Advanced Mathematics by Smith"?

The main purpose of this book is to provide a bridge between introductory mathematics courses and more advanced topics, helping students develop the necessary skills and understanding to succeed in higher level mathematics courses.

2. Is this book suitable for self-study or is it better used in a classroom setting?

This book can be used for both self-study and in a classroom setting. It is designed to be approachable for independent study, but also includes exercises and problems for use in a classroom setting.

3. What topics are covered in this book?

This book covers a wide range of topics in advanced mathematics, including logic, proof techniques, set theory, functions, relations, and basic number theory. It also introduces students to more abstract mathematical structures such as groups, rings, and fields.

4. Is this book appropriate for all levels of students?

This book is primarily aimed at students who have completed an introductory course in mathematics, such as calculus, and are ready to move on to more advanced topics. However, it can also be helpful for students who are just starting to explore higher level mathematics.

5. Are there any supplemental resources available for this book?

Yes, there are several resources available to supplement this book, including online problem sets, study guides, and additional practice exercises. These can be found on the publisher's website or through the instructor's materials.

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