Cross-post from University of Reddit:
Introduction:
This course is designed to act as an introduction to proof based high-level mathematics. It guides students to think mathematically by analyzing a problem, extracting the pertinent information, and drawing appropriate conclusions.
This course will follow “A Transition to Advanced Mathematics” (Smith, Eggen, St. Andre; 6th Edition).
“In summary, out main goals in this text are to improve the student’s ability to think and write in a mature mathematical fashion and to provide a solid understanding of the material most useful for advanced coursed… in almost any mathematically related work you may do, the kind of reasoning you need to be able to do is the same reasoning you sue in proving theorems.” (Smith, preface)
I believe that understanding mathematics is a prerequisite to all other understanding. I hope that through this course I can convince you of the validity of this statement.
Prerequisites:
None. Really. This is the beauty of mathematics. Everything is accessible through time and devotion. This course will start with simple logic and axioms and build up to more complex ideas.
Content:
This course will cover the following topics (subject to change):
- Logic and Proofs
- Propositions and Connectives
- Conditionals and Bi-conditionals
- Quantifiers -Basic Proof Methods
- Proofs Involving Quantifiers
- Additional Examples of Proofs
- Set Theory
- Basic Concepts of Set Theory
- Set Operations
- Extended Set Operations and Indexed Families of Sets
- Induction -Equivalent Forms of Induction
- Principles of Counting
- Relations
- Cartesian Products and Relations
- Equivalence Relations
- Partitions
- Ordering Relations -Graphs
- Functions
- Functions as Relations
- Constructions of Functions
- Surjective (Onto); Injective (One-to-One)
- Image of Sets -Sequences
I do not intend to cover the last two sections of the book titled “Concepts of Algebra” and “Concepts of Analysis”. I will cover this topics in later courses.
The lessons will be posted on my new website: http://thisismath.org. The classes will start later this week, and I am considering my options for live chat.
Additional Information
“If I have seen further than others, it is by standing upon the shoulders of giants.” -Newton
Mathematics has an unmatched ability to bring the brightest minds of the world to humility.
I am not the best mathematician. I don’t even consider myself a good mathematician. Frankly, my computational skills are par at best. Indeed there exists a countable set of individuals that would be multitudes better to teach this material than myself. However, I am unmatched in my passion. I am driven to convince people that they need to experience the beauty of advanced maths and the mode of thinking it forces upon them. Because of this, I will put forth an extraordinary level of effort to make advanced maths available to anyone willing to put in the time.
“high academic achievers are not necessarily born ‘smarter’ than other, but work harder and develop more self-discipline” -David Shenk, ‘The Genius in All of Us.
Work hard and diciplined and I promise to teach you all that I know about Mathematics.
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