Lecture Notes for transition to advanced has contents: Logic and proof, set theory and induction, relations, functions, cardnality, modular arithmetic, algebra, basic logical operations, definitions for our toy examples | Lecture Notes for Transition to Advanced Mathematics James S. Cook Liberty University Department of Mathematics and Physics Spring 2009 1 introduction and motivations for these notes These notes are intended to complement your text. I intend to collect all the most important definitions and examples from your text. Additionally, I may add a few examples of my own. I plan to organize these notes lecture by lecture. This semester we should have a total of 23 lectures. Each lecture should last approximately one-hour. Approximately once a week I will take 10 or so minutes to tell you a bit of math history and development, we’ll trace out some of the major themes and directions which modern mathematical research includes. As usual there are many things in lecture which you will not really understand until later. Completing homework in a timely manner is a crucial step in the learning process. Also it may be necessary to read the text more than once to really get it. One size does not fit all, it may be that you need to do more homework than I assign in order to really comprehend the material. I may give a quiz from time to time to help you assess your level of understanding. In some sense this a survey course. It’s a mile wide and an inch deep. However, there are certain issues we will take very seriously in this course. It should be fairly obvious from the homework what is and is not important. I hope you will leave this course with a working knowledge of: conditional and biconditional proofs proof by contradiction proof by contraposition proof by the principle of mathematical induction proper use of set notation and mathematical short-hand given sets A, B, how to prove A ⊆ B given sets A, B, how to prove A = B equivalence classes and relations proving a function is injective (1-1), surjective(onto) proving a function is bijective(1-1 and onto) 1 basic terminologies in abstract algebra modular arithmetic (how to add subtract and divide in Zm ) Finally, I hope you .
