Math 265: Transition to Advanced Math

Meeting Information

Instructor: Lauren Williams
Meeting Times: MWF 9:15-10:20 AM
Meeting Location: Hirt M209
Office: Old Main 404 (Tower)
Office Hours: Mon 10:45-12 and 4-5, Tues 9:15-12, Wed 10:45-12, Fri 10:45-12 and by appointment

Course Objectives

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 the book A Gentle Introduction to the Art of Mathematics, Version 3.1SN, by Joe Fields. You will not need any additional materials for the course.

The author of this book has generously made it available as a free pdf. You can also order a printed copy from CreateSpace for $16 if you prefer, but this is not required for the course.

There is also a workbook to accompany the text if you're looking for extra practice.


You will have homework assignments due approximately every week. These assignments will feature several questions taken from the text as well as other sources. Using proper mathematical notation and language will be a major part of your grade, aside from simply answering questions correctly. You will also be given additional suggested problems that will not be collected, but will help give you sufficient practice to succeed in the course. You are permitted and encouraged to work together on all assignments. Your lowest homework grade will be dropped when calculating your grade.

Homework assignments will be posted here, as well as distributed in class. Solutions will be posted shortly after assignments have been collected.

Due Date Assignment Solutions
September 5
September 10
September 17
September 24
October 1
October 15
October 22
October 29
November 5
November 19
December 3


We will have two in class exams on the following dates. You will be given an exact list of topics, along with a review sheet posted here, approximately one week before each exam. Use of notes, textbooks, calculators, electronic devices, or other materials will not be permitted during an exam.

Your lowest exam grade will be replaced by your final exam grade, if your final exam grade is better. There are no make up exams; a missed exam grade will be replaced by your final exam grade. A second missed exam will receive a grade of 0, so please check your schedules carefully and ensure that you can attend all exams.
  • Monday, October 6
  • Monday, November 10

Exam review sheets will be posted here approximately one week before each exam:

Review Sheet Solutions
Exam I Coming Soon (will be posted by September 29) Coming Soon (will be posted by October 4)
Exam II Coming Soon (will be posted by November 3) Coming Soon (will be posted by November 8)

Final Exam

The final exam will be cumulative, and is scheduled for Friday, December 12, 8:30 - 10:30.

Final Grades

Your final grade will be calculated as follows:
  • 30% - Average of 2 in class exams
  • 50% - Average of homework assignments
  • 20% - Final exam
Grading scale:
F D D+ C C+ B B+ A
0-59 60-64 65-69 70-77 78-83 84-89 90-93 94-100
Quiz and exam grades will be posted on Blackboard, so you can keep track of your progress at any time.

Other Course Information

  • You are neither expected nor required to purchase any materials for the course aside from the required textbook. Graphing calculators and mathematical software could be used to check your work, but should not be relied on to do the work for you.
  • I will attempt to return emails as thoroughly and promptly as possible. However, it is generally better to ask complicated questions during class or in office hours. If you have a question about the homework, it is quite likely someone else has the same question, so you're doing the class a favor by asking!
  • I do not keep detailed lecture notes. It is highly recommended that you establish contacts among your classmates to get notes in case you miss class.
  • Attendance is not required, but coming to class regularly will generally improve your grade. You are responsible for any work material covered in your absence. Please contact me if you are absent for an extended period.

Mathematics Department Student Learning Outcomes

Your written homework in this course will be used to assess your ability to effectively write mathematics. This assessment does not affect your grade, and a separate rubric will be used for the assessment vs your assignment grade.

Learning Differences

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.

Support of the Mercy Mission

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.

Extra Resources

Cut the Knot's Proof Page
A nice list of some simple but famous proofs, along with some fallacies.

Wolfram Alpha
If you're not already familiar with it, Wolfram's (free!) online math software is a great reference. Aside from performing calculations and drawing graphs, you can ask Alpha to provide definitions, explanations, and examples of many of the topics we'll see in class.

An open source mathematics software system. Runs natively on Linux and Mac, but you can also run it within your browser. Plenty of documentation to help offset the learning curve. Based on Python with plenty of useful packages, and you can contribute!

Course Schedule

The schedule below is approximate - topics covered on a particular day are subject to change. Exams and quizzes will take place as scheduled, with adjustments to material covered made when necessary. Any changes to material covered on quizzes will be announced in class and updated here.

Aug 27 Class Introduction, Types of numbers
Aug 29 Basic Number Theory, Relations
Sep 1 No Class (Labor Day)
Sep 3 Section 2.1 Predicates and Logical Connectives
Homework 1 Due
Sep 5 Section 2.2 Implication
Sep 8 Section 2.3 Logical Equivalences
Sep 10 Section 2.4 Two Column Proofs
Homework 2 Due
Sep 12 Section 2.5 Quantified Statements
Sep 15 Section 2.6 Deductive Reasoning and Argument Forms
Sep 17 Section 2.7 Validity of Arguments and Common Errors
Homework 3 Due
Sep 19 Section 3.1 Direct Proofs of Universal Statements
Sep 22 Section 3.2 More Direct Proofs
Sep 24 Section 3.3 Contradiction and Contraposition
Homework 4 Due
Sep 26 Section 3.4 Disproofs
Sep 29 Section 3.5 By Cases and By Exhaustion
Oct 1 Section 3.6 Existential Statements
Homework 5 Due
Oct 3 Review for Exam I
Oct 6 Exam I
Oct 8 Section 4.1 Basic Notions of Set Theory
Oct 10 No Class (Mid Semester Break)
Oct 13 Section 4.2 Containment
Oct 15 Section 4.3 Set Operations
Homework 6 Due
Oct 17 Section 4.4 Venn Diagrams
Oct 20 Section 5.1 The Principal of Mathematical Induction
Oct 22 Section 5.2 Formulas for Sums and Products
Homework 7 Due
Oct 24 Section 5.3 Other Proofs Using PMI
Oct 27 Section 6.1 Relations
Oct 29 Section 6.2 Properties of Relations
Homework 8 Due
Oct 31 Section 6.3 Equivalence Relations
Nov 3 Section 6.4 Ordering Relations
Nov 5 Section 6.5 Functions
Section 6.6 Special Functions
Homework 9 Due
Nov 7 Review for Exam II
Nov 10 Exam II
Nov 12 Section 7.1 Counting
Nov 14 Section 7.2 Parity and Counting Arguments
Nov 17 Section 7.3 The Pigeonhole Principle
Section 7.4 The Algebra of Combinations
Nov 19 Section 8.1 Equivalent Sets
Homework 10 Due
Nov 21 Section 8.2 Examples of Set Equivalence
Nov 24-28 No Class (Thanksgiving Break)
Dec 1 Section 8.3 Cantor's Theorem
Dec 3 Selected Topics
Homework 11 Due
Dec 5 Review for Final Exam
Dec 12 Final Exam 8:30 - 10:30