Math 265 Transition to Advanced Mathematics

Meeting Times: Monday, Wednesday, and Friday, 12:00 - 12:50 pm
Location: Hirt 209
Office Hours: Monday 9-10, Tuesday 12:30-1:30, Wednesday 1-3, Thursday 8-9, Friday 1-2
Prerequisite: Math 170, Math 150

This course is designed to facilitate the mathematics student’s transition to courses requiring a higher level of mathematical maturity. Emphasis will be on the reading and writing of proofs, and on communicating mathematically—both orally and in writing. Topics will include logic, set theory, functions, relations, and number theory.

In this course, you will:

• learn to write using formal, mathematical language with correct notation.
• learn to construct direct proofs, proof by contradiction, and proofs by induction.
• learn to read mathematics critically, and be able to determine whether a proof is sound or flawed.
• define relations between sets of objects and the properties of those relations.
• learn the basic definitions and principles of logic, set theory, combinatorics, and number theory.
• be exposed to several different areas of mathematics, via direct study or within examples designed to clarify other topics.
• learn to apply new techniques of problem solving to challenging material, both in this course and in future study.

We will be using Book of Proof, Second Edition, by Richard Hammack. You can download the entirely textbook for free from the link, or order an inexpensive printed copy from an online bookseller or the Mercyhurst bookstore.

This course is intended for students pursuing a major or minor in mathematics. Calculus I and Linear Algebra, or instructor permission, are required.

There will be 10 homework assignments throughout the semester, typically due one week after they are assigned. Some problems will be marked "suggested" and will not be collected. These will usually be problems from the textbook; solutions to these problems are available on the website for the textbook. These problems may appear on exams, so you are strongly recommended to work through them.

For collected problems, you are expected to submit your final work. Problems involving calculations should include work and an explanation of the steps used to arrive at your answer. Proofs should use formal language and notation, as covered in class. Work should be clear and neatly written.

You should expect to spend a fair amount of time on each assignment - don't wait until the night before it's due to get started! You are free to work together on your assignments, but everyone must submit their own work, in their own words. Late homework will be accepted with a 10% per day penalty, until the graded assignment is returned to class.

If you are unable to submit your work in class, you can email a clear scan or photo of your work.

We will have two midterm exams and a final exam. The final exam will be cumulative, while the midterm exams will focus on more recent material. Both exams will be based on homework problems and the suggested textbook problems that do not need to be turned in.

You will be able to make up exams for excused absences. If you know in advance that you will not be able to take an exam at its schedule time, please let me know as soon as possible. All make ups must be completed within one week of the exam date.

The final exam is cumulative, including material from all sections covered in class.} Most questions on the final will be taken (with minor modifications) from homework and previous exams.

You are required to take the final exam for this course regardless of your average on earlier exams or quizzes. If you will not be able to take the final exam at its scheduled time, please make alternate arrangements as soon as possible. Final exams may be made up for excused absences only.

Your final grade will be calculated as follows:

160 points	Midterm Exams
	Two exams, 80 points each
240 points	Homework
	Ten take home assignments, 24 points each
100 points	Final Exam

Your letter grade will be determined according to the department grading scale:

In keeping with college policy, any student with a disability who needs academic accommodations must call Learning Differences Program secretary at 824-3017, to arrange a confidential appointment with the director of the Learning Differences Program during the first week of classes.

This course supports the mission of Mercyhurst University by creating students who are intellectually creative. Students will foster this creativity by: applying critical thinking and qualitative reasoning techniques to new disciplines; developing, analyzing, and synthesizing scientific ideas; and engaging in innovative problem solving strategies.

Free Books

A Gentle Introduction to the Art of Mathematics, Joe Fields
Free textbook: "GIAM (a Gentle Introduction to the Art of Mathematics) is a free, open-source textbook -- the current version is 3.1. GIAM covers several topics in the foundations of mathematics (logic, sets, relations, functions and cardinality) and introduces the reader to many techniques of mathematical proof (direct, indirect, contradiction, contrapositive, mathematical induction, combinatorial proofs and magic). There are amusing quotations at the start of each chapter."

Linear Algebra, Jim Hefferon
Linear algebra textbook, if you need a refresher on systems of equations, vector spaces, etc.

Free Software

Wolfram Alpha (Web Application)
Use it to check your work and visualize graphs. From the makers of Mathematica. A (modestly priced) upgrade is available, but the free version allows unlimited computations without an account.

GAP (Unix, Mac OS, or Windows)
"GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. See also the overview and the description of the mathematical capabilities. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. The system, including source, is distributed freely. You can study and easily modify or extend it for your special use."

CoCalc (Web Application)
An online computing system that includes SageMath, a powerful mathematics software system based on Python. CoCalc also includes GAP, Maxima, SymPy, R, Julia, Octave, and even LaTeX. An easy way to try them all out without installing any software! Plenty of documentation to help offset the learning curve.