CARROLL COLLEGE

Department of Mathematics , Engineering, Physics,

and Computer Science

MA 301     Foundations of Mathematics - Spring, 2001 

11 - 11:50 a.m. MWF  123 SI                                 

Instructor:  Philip B. Rose                 

Text:    Foundations of Higher Mathematics   3^rd Edition by Fletcher and Patty, PWS Publishers

Description:  This course is intended to be an introduction to mathematical rigor and proof.  A specific goal is to help prepare students for the demands of standard junior-senior courses in analysis and algebra.  The approach is strictly hands-on.  In this class we concentrate on the "language" that all mathematicians use, namely, proofs.  This will strengthen our ability to understand and communicate mathematics in all its various forms.  More specifically, we will study basic techniques for formulating and writing mathematical proofs.  Throughout the course complete proofs are given that illustrate proof techniques and are intended to serve as models.  The course is organized in a manner which parallels the content of the text.  We cannot discuss every specific topic in the text, but the course is designed so that your understanding of the material depends heavily on studying the text and doing the drill problems and outside assignments.  We start with a review of logic and the language of proofs, and then a review of sets.  We then study types of proofs including mathematical induction, relations and orders, functions, a little combinatorics, a look at countable and uncountable sets, and finish with topics chosen from introductory group theory and the foundations of calculus.    At this writing, there are a series of 'labs' to go with this class.  These are extensive documents in Mathematica which are designed to help you learn some mathematics by doing mathematics.  I’ll discuss these in detail when we get to the first one.  There will be homework every class period; these will frequently be collected and graded. 

Grading:  homework, projects, labs:  40%    3 hour exams: 40%    final: 20%

Office Hours:  MW: 9-11 a.m., 1-2 p.m., 3-4 p.m.;  TuTh: 9-11 a.m., 1-4 p.m.;  F: 9-11 a.m., 1-2 p.m.

Comments:  This can be a very rewarding class, but it will require hard and steady work.  Try not to miss class.  Although we have an excellent text we will be looking at much material that is not in the text.  Try not to miss class. If you do miss class, it is your responsibility to get the notes from someone else, and only then come in to see me if you have questions. Regarding working together, I not only do not mind your discussing questions, problems, discoveries, etc. with your colleagues, but encourage it  But I do want to see your own work on assignments.  The schedule on the next page is tentative, and is to be seen as a general guide. I will always announce exams at least a week in advance.

FINAL EXAM:  Monday, May 7, 2001,   10 a.m. - noon.  Please make a note of this now.  This date and time may not be moved so make your end-of-school travel and other plans accordingly.      

Tentative Course Outline

JAN  17          W    Introduction, propositions, expressions, and tautologies

          19          F      Quantifiers, methods of proof

          22          M     Proof by contradiction and other proofs

          24          W    Introduction to Sets, operations on sets

          26          F      Indexed families, an axiomatic approach to sets

          29          M     Proofs by Induction

          31          W    Other principles of induction

FEB     2          F      Induction and Recursion

            5          M     Consequences of the Division Algorithm

            7          W    A Review of some Number Theory

            9          F      Quiz 1 over chapters 1, 2, 3

           12         M     Relations

          14          W    Cartesian Graphs and Directed Graphs

          16          F      Equivalence Relations

          19          M     PRESIDENTS DAY - NO CLASSES

          21          W    Partitions and Identifications

          23          F      Congruence

          26          M     Composition of Relations

          28          W    Types of Orders

MAR  2          F      Functions as Relations

            5          M     Functions viewed Globally

            7          W    Permutations

            9          F      Functions and Partitions

      12-16         M-F SPRING BREAK

          19          M     Real-valued Functions

          21          W    Images and Inverse Images of Sets

          23          F      Functions and indexed Families

          26          M     Quiz 2 over chapters 4, 5

          28          W    The Sum and Product Rules

          30          F      Dirichlet's Pigeonhole Principle

       APR           2            M         The Binomial Theorem

              4          W    Graphs

             6         F      Finite and Infinite Sets

            9          M     The Schroeder-Bernstein Theorem

         11          W    The Schroeder-Bernstein    Theorem

         13          F      Good Friday, no classes

          16          M     Easter Monday, no classes

          18          W    The Well-Ordering Principle and the Axiom of Choice, Countable Sets

          20         F      Uncountable Sets

          23          M     More on Countable and Uncountable Sets                                                                

          25          W    Cardinal Arithmetic

          27          F      Quiz 3 over chapters 6, 7

          30          M     Sequences and Convergence

MAY     2          W    Subsequences and Arithmetic Operations on Sequences

            4          F      final topics and review

              5, 7 - 10     FINAL EXAMS

FINAL EXAM:  Monday, May 7, 2001,   10 a.m. - noon.