Foundations A Transition to Advanced Mathematics by Smith

[micromass]

• Author: Douglas Smith, Maurice Eggen, Richard St. Andre
• Title: A Transition to Advanced Mathematics
• Amazon Link: https://www.amazon.com/dp/0495562025/?tag=pfamazon01-20
• Level: Undergrad

Table of Contents:

• 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
  • Additional Examples of Proofs
• 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
• Answers to Selected Exercises
• Index

 

[dkotschessaa]

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.

 

[micromass]

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!

 

[dkotschessaa]

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.

 

[theoristo]

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]

I also found one legally free book online http://www.people.vcu.edu/~rhammack/BookOfProof/index.html
which has good review on amazon https://www.amazon.com/dp/0982406207/?tag=pfamazon01-20

 

[dkotschessaa]

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.

If anything, they can complement each other well. I'd be glad to talk more about this on another thread since this one is focused on Smith.

-Dave K

 

[theoristo]

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.

If anything, they can complement each other well. I'd be glad to talk more about this on another thread since this one is focused on Smith.

-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.

 

[dkotschessaa]

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

 

[theoristo]

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?

 

[dkotschessaa]

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.

 

[lurflurf]

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.

 

[dkotschessaa]

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

 

[lurflurf]

^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.

The best thing about this book is that if one has read it they will know what some of these mean
$$\aleph \in \ni \neg \sim \mathbb{NZRC} \Rightarrow \Leftrightarrow \wedge \vee \bot \forall \exists \equiv $$
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.