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Foundations Intro to Proof (Foundational) Mathematics

  1. Feb 3, 2016 #1
    Hello,

    I'm trying to self teach myself Fundamental Mathematics. I looked around, but I wasn't sure what to look for exactly. I read the part on Set Theory in "Book of Proof" by Richard Hammack. I enjoyed it, but I wasn't sure if it is rigorous enough to stand against a college level course.

    The course I'm trying to mimic goes by this:
    • Chapter 1: Numbers, Sets, and Functions (2 - 4 hours)
      • Set-theoretic concepts and operations
      • Functions: Formal definition and basic concepts (graph, image set, bounded function, decreasing/increasing function)
      • Triangle and AGM inequalities
      Note: Much of the material in Chapter 1 can be covered at a later point. For example, one could spend one or two hours on basic set-theoretic concepts, then move on to logical statements in Chapter 2, while deferring the formal definition of functions and related concepts to the beginning of Chapter 4.
    • Chapter 2: Language and Proof (4 - 6 hours)
      • Logical statements, conditionals, quantifiers
      • Methods of proof (direct, contraposition, contradiction)
    • Chapter 3: Induction (4 - 6 hours)
      • Sum/product notations
      • Induction and strong induction
      • Well-ordering principle
      • Applications
    • Chapter 4: Bijections and Cardinality (4 - 6 hours)
      • More about functions: injective, surjective, bijective properties, compositions and inverses
      • Cardinality, finite, countable, and uncountable sets
      • Countability of rationals
    • Chapters 13/14: Real Numbers, Sequences and Series (8 - 10 hours)
      • Infinite sequences and series: Formal definition of convergence, basic properties of convergent sequences and series
      • Sup, inf, and the Completeness Axiom
      • Cauchy sequences and the Cauchy Convergence Criterion
      • Montone Convergence Theorem
      • Bolzano-Weierstrass Theorem
      Note: These chapters introduce students to basic concepts in analysis and form an essential component of this course.

    Obviously I could just their text, D'Angelo/West, Mathematical Thinking, Second Edition, but generally books recommended for classes aren't the best for self taught methods.

    tl;dr best book for self teaching fundamental mathematics (Course of Topics above)
     
  2. jcsd
  3. Feb 6, 2016 #2

    QuantumQuest

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    It clearly depends on what is your background and what exactly you want to pursue. Your background will tell how easily or not, fast or not and in what order/ combinations you can take a course like the one you describe and what resources you would need. So, it would be necessary to tell something about your background. Moreover your goal is equally important. Is there something that you want to pursue further and what is it? There is a multitude of resources on the web you can effectively use, like Wikipedia, online tutorials, notes from various universities and free online courses (Coursera, edx to name a few), so your choices besides textbooks, are really many.
     
  4. Feb 6, 2016 #3
    I'm currently taking Multivariable calculus. I've done some Set Theory through Book of Proof by Richard Hammack. I've also had a Honors Calculus 2 class, which was more theoretical.

    My goals are to have a good base of proof based mathematics or an intro to "real math"

    I'm trying to prepare myself for a theory based Differential Equations course and just get an overview of proof based mathematics. Below is the syllabus for said Diff. Eqn. class.

    Introduction
    First order linear equations
    Separable and exact equations
    Integrating factors and
    homogeneous equations
    Differences between linear and nonlinear equations
    Bernoulli Equations
    Modeling with first order equations
    Autonomous equations and population dynamics
    The Picard existence and uniqueness theorem
    The basic theory or nth order linear differential equations
    nth order linear equations with constant coefficients
    Undetermined coefficients and variation of parameters for nth order equations
    Applications
    Review of power series
    Series solutions near an ordinary point
    Regular singularities, Euler equations
    Series solutions near a regular singular point (I, II)
    Bessel's equation
    Introduction to systems
    Solving linear systems with constant coefficients
    Stability and Liapunov's Method

    Lastly, I'm currently looking at Numbers and Functions: Steps into Analysis by R. P. Burns. Would you recommend this book for what I'm trying to learn? If not what would you recommend?
     
  5. Feb 6, 2016 #4
    Try any discrete matchbook. I like Epps as a gentle introduction to proofs.
     
  6. Feb 6, 2016 #5
    Aren't those books more of directed at programmers? I was going to start out with Numbers and Functions: Steps into Analysis by R. P. Burns or Understanding Analysis. Would that book be accessible with Calculus under my belt?

    I figured the Analysis book is good middleground, but I'm not sure if I'll be able to understand it
     
    Last edited: Feb 6, 2016
  7. Feb 6, 2016 #6
    Some Discrete Math books are aimed at the computer science curriculum, however, Epps is not. Discrete math Course is typically a collection of math topics thrown into one course. Discrete Math Course (intro) has the following: first part of the semester acquaints students with how to read and write proofs, you see your number theory, graph theory, combinatorics, probability etc. There can be more or less material depending at the college/instructor taken.

    Discrete math serves as bridge to logical thinking and higher mathematics. If your differential equations course focuses more on theory, I would wait until you have completed Linear Algebra and possibly Cal 3.

    Not sure of Analysis. I am reading Spivak currently and going to go through Shilov's Analysis right after. Spivak is really noob friendly. If you have not written proofs before, I would start building mathematical maturity with Discrete Math/ Linear Algebra by doing all the proofs and most exercises from a good book.

    I am not the most experienced person here, I am a student, so ask others. Micromass offers very good information.
     
  8. Feb 7, 2016 #7

    QuantumQuest

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    So, as I see, you want to go in a theoretical math path. Having taken a Functional Analysis course in the past - besides the usual Calculus I,II etc., I would recommend to get into Topology and Functional Analysis when you're ready for that - i.e. after finishing a Calculus III class and Differential Equations. A good book in my opinion is Munkres' "Topology" and Kolmogorov - Fomin "Elements of the theory of functions and functional analysis" and "Functional Analysis" by H.Brezis. I would also recommend "Introductory Real Analysis" by Kolmogorov, if you want to get first in an introduction to Real Analysis. I have used myself the aforementioned books and some parts of them on a self - study basis.
     
  9. Feb 7, 2016 #8

    Fredrik

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    It is. Its level of rigor is appropriate for courses in algebra and analysis.
     
  10. Feb 7, 2016 #9
    So would it be advisable to keep going through Book of Proof and Understanding Analysis at the same time or Book of Proof first? With Problems in Real Analysis as a supplement?

    I've redefined my goals through the progression of the thread. I would like to follow my dream of learning Analysis, but I assume you need a good base of logic and proof writing for it.
     
  11. Feb 7, 2016 #10
    Well, i've tried Abbott - Understanding Analysis and Rudin - PMA, both didn't go too well to ne, in the former there's a extreme lack of continuity and rigor. i.e: In the first chapter he tried to prove that sqrt(2) is irrational, what he explains: that for a real number be rational it has to be the ratio of 2 intergers and made a proof by exposing that for exist a a number that is the square root of 2, the intergers have to be prime(euclides' lemma), which doesn't happen in this case, yes it's beautiful proof, but don't give any insight about the structure of the proof, and in the exercises he asked to proof that 3 and 6 is irrational, and why 4 isn't, i gived up from the book because the lack of clarity. Then i discovered Bloch - Real Numbers and Real Analysis, which since beggining showed clarity and rigor in the proofs, by PROVING and constructing the intergers, rationals and reals, which he begins by the Peano Postulates in the naturals, and in chapter 2 he proves that every number in the Positive Intergers has a root that lies in the Reals and that exists some numbers that the root lies in the Irrationals. A simple beautiful book, try him if you have a chance ;)
     
  12. Feb 7, 2016 #11

    micromass

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    I would keep going with the Book of Proof. It's a decent book. You don't need any further book. You can probably read the Book of Proof and Understanding Analysis concurrently.

    if you really want alternatives, then I like both of Bloch's books: one is a substitute for the Book of Proof, the other is an excellent real analysis text:
    https://www.amazon.com/Proofs-Fundamentals-Abstract-Mathematics-Undergraduate/dp/1441971262
    https://www.amazon.com/The-Real-Numbers-Analysis/dp/0387721762
     
    Last edited by a moderator: May 7, 2017
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