Math 300: Fundamental Concepts of Mathematics

T, Th 2:30-3:45 in LGRT 206

Instructor:
Dr. Liubomir Chiriac, 1115J LGRT (email: mylastname [at] math [dot] umass [dot] edu).

Office Hours:
Wed 3-4, or by appointment.

Course Description:
This course is an introduction to rigorous abstract mathematics. In lower-level classes like calculus, the emphasis is on applying formulas and theorems to specific problems. In this class, we will be more concerned with why the formulas and theorems are true. In mathematics, the way we know that a statement is true is by giving a proof of it. We will learn what a proof is, how to read, create, and present proofs, and how to tell a correct proof from an incorrect one.

Textbook:
  • Mathematical Reasoning: Writing and Proof by Ted Sundstrom.
  • A free copy of the book can be downloaded here. The option of buying a printed version is also available.


  • List of topics:
    Propositional logic and quantifiers. Methods of proof: direct proofs, proof by contradiction, proof by induction. Elementary set theory. Functions: injections, surjections, bijections, inverse functions. Equivalence relations and equivalence classes. Elementary number theory: divisibility, congruence, greatest common divisor, the Euclidean algorithm. Cardinality: finite sets, countable sets, uncountable sets.

    Grading:
  • Homework and class participation - 30%;
  • Two Midterms - 20% (each);
  • Final - 30%.

  • Grading scale:

    90-100

    85-89

    81-84

    77-80

    73-76

    69-72

    65-68

    61-64

    56-60

    51-55

    0-50

    A

    A-

    B+

    B

    B-

    C+

    C

    C-

    D+

    D

    F


    Exams:
  • Midterm 1 - Thursday, 2/21/19, in-class. Covered material: Chapter 1, 2 and 3.
  • Midterm 2 - Thursday, 4/4/19, in-class. Covered material: Chapter 4, 5 (including Binomial Coefficients) and 6.
  • Final - Friday, 5/3/19, 3:30-5:30, in LGRT 121. The emphasis will be on Chapters 7,8, and 9.

  • Below is a list of practice problems for the final (with solutions in the textbook):
    7.3:3,5(a),6(a) 7.4:2(a,e,g),5(a) 8.1:5(a,b,e),6(a) 8.3:3(a,b) 9.1:3 9.2:6 9.3:4.

    Homework:
  • Homework will be due on Thursdays at the start of lecture, unless otherwise stated.
  • Assignments turned in by the next Tuesday will receive 75% credit. No assignments will be accepted after that.
  • Note that you will need to do the reading to be able to do the homework. If you're stuck, it's ok to consult your classmates or me. However, the work you turn in must be entirely your own, in your own words!
  • All Exercises are from the textbook. Note that "x.y:N" is to be read Exercise N from Section x.y.

  • Problem Set 1 (Due 01/31): 1.1:6(d,e,f) 1.2:4(c),7,10(bcd),12 2.1:4(a,b,d),6,12(a)
    Problem Set 2 (Due 02/07): 2.2:5,8,9(a,b,c) 2.3:4(c,d) 2.4:2(c,e),4(b,f),11
    Problem Set 3 (Due 02/14): 3.1:10,11(c) 3.2:5,17(d,e) 3.3:2(c) 3.4:6(b)
    Problem Set 4 (Due 02/21): Practice Midterm I
    Problem Set 5 (Due 03/07): 4.1:3(c),8(b) 4.2:10,11 4.3:2(d),11
    Problem Set 6 (Due 03/21): HW 6
    Problem Set 7 (Due 03/28): 5.2:14 5.3:3,6(b),11(a) 5.4:7 6.1:7(d,e,f,g,h)
    Problem Set 8 (Due 04/04): Practice Midterm II
    Problem Set 9 (Due 04/18): 8.1:3,5(c,d,f),6(b,c) 8.2:8,12 8.3:3(e,h),8,9
    Problem Set 10 (Due 04/25): 7.2:5,13 7.3:5(b),8 7.4:2(d,f),6,8,19.

    Attendance:
    Attendance is required during lectures. I consider attendance and participation important ingredients for your success in the course. Frequent absences will be reflected in your grade.

    Make-up Exam Policy:
    If you have a legitimate (e.g. multiple exams at the same exact time, medical problems, emergency absences, religious observances) scheduling conflict with any of the exams, it is your responsibility to notify me of the conflict at least two weeks in advance. Please note that previously arranged travel plans are not a valid reason to be given a make-up exam.