MTH 311 Introduction to Higher Mathematics Dr. Adam S. Sikora, Dr. Joseph A. Hundley, and Dr. Michael J. Cowen

c 2020 by Adam Sikora, Joseph A.Hundley, and Michael J. Cowen.

Version April 9, 2021 2

Preface The first objective of this course is to teach students the skills of precision in thinking, proving theorems, and of communicating math- ematics using precise language and proper notation. The second objective is to introduce students into higher mathe- matics. After discussing logic, sets, and functions, these notes follow the traditional development of numbers, starting with Peano’s Axioms for the naturals and then successive definitions of integers, rationals, reals, and complex numbers. Students learn a variety of topics along the way: recursive sequences including Fibonacci numbers, equivalence relations, rudimentary , including Fermat’s Little Theo- rem, and Russell’s Paradox. Some basic concepts of abstract algebra are introduced, including the notions of rings and fields. Finally, countability of a set is discussed, including uncountability of the reals, the notion of cardinal numbers, Cantor’s Hypothesis, and its independence from Peano’s axioms. A significant number of proofs and examples are included in these notes.

How this course differs from others This course differs from other courses you have taken in three ways: 1. In calculus and other elementary math courses you were exposed to a variety of routine mathematical problems and methods were introduced for solving each of them. After each class, you were asked to practice problems following the examples explained in detail in class. There will only a LIMITED amount of that approach in this course. Often you will be asked to think about less routine problems with NO practice solutions. As there is no universal recipe for proof writing, that skill will be difficult for some of you to acquire. 2. The course covers the most abstract mathematics you have ever encountered. Sometimes, it may be hard to “draw a picture” or give a simple example to illustrate a new concept. 3. In lower level math courses, your focus was on finding the correct answer to a problem, usually a number or a function, which would appear at the bottom of your solution. The rest of your solution, leading to the desired answer, might have looked like a scratch work. That kind of solution work will NOT be satisfactory here. This course intends to teach a proper, clear style of mathematical communi- cation. Therefore, each problem solution has to be presented in such a way that others can read it and understand it. NO scratch style work 3 will be accepted. You cannot start your answer with, say, “y > 1”. If you do, I will ask you what is y? Are you assuming y > 1? (Why?) or concluding that from some other assumptions? (Also why?) Needless to say, all solutions have to be written in proper English. In the future, you will likely want or need to explain some mathematical concept or reasoning to others. Since math is a challenging subject, it can’t be explained with messy notation, undefined symbols and stream- of-consciousness style of writing. Leave that to post-modern poets. We will discuss further details on solution writing and proof writing in Section3.

How to do well. Come to class and recitation. Study these notes beforehand. Mathematics requires PRECISION. A implies B does not mean that B implies A. Make sure you have a precise understanding of the definitions and theorems. “Sort-of” understanding is not enough in this course. If you are asked to compute the range of a certain function, then you have to know precisely what the range of a function means. Ask questions on the material you do not understand. Learn the definitions and statements of propositions (lemmas, theo- rems,..) thoroughly. • Memorize each definition and proposition. • Go over each definition, giving examples and non-examples, un- til you understand the idea behind the definition and c