Introduction to Advanced Mathematics, MATH 215, Spring 2017

Instructor: Daniel Groves, 727 SEO e.mail

Course webpage: http://www.math.uic.edu/~groves/teaching/2016-17/215/

Syllabus: Download here.

Course hours:

MWF, 1:00-1:50PM, Room B10, Burnham Hall.

Office hours:

Mondays 11am, Fridays 10am.

(Or, you can make an apointment by e.mail or try stopping past my office.)

Text: "An Introduction to Mathematical Reasoning", by P. Eccles, Cambridge University Press. ISBN:9780521597180


Course description:
The goal of this course is to learn how to create and write mathematical proofs, and to learn why one might want to do such a thing. We will introduce and study some important mathematical concepts used in advanced mathematics courses, particularly equivalence relations.

Assessment:
There will be homework for most classes, two in class midterm exams and a final exam. Since there will be a lot of writing, explaining and critiquing in class, there will also be a class participation component of the grade. The relative weighting of these components will be:
  • Homework: 20%
  • Class participation: 10%
  • Midterm exams: 20% each
  • Final exam: 30%

    Exams:

    Two midterms, and one final, dates and times to be announced.

  • The first midterm will be on Friday, February 17 during class.

    Here are some worked solutions to the first midterm. For reference, here is the handout from the midterm. (The actual exam is included as part of the solutions.)

  • The second midterm will be on Friday, March 31 during class.

    Here are some worked solutions to the second midterm. For reference, here is the handout from the second midterm.

    Daily Homework:

  • For Wednesday, January 11, 2017, in class: In class on Monday, 1/9, we worked on this worksheet. For class on Wednesday, prove Proposition 4. Decide whether or not Conjecture 5 is true. If it is true, prove it. If it is not, prove it is not.

  • For Friday, January 13, 2017, in class: Prove Propositions 6, 7, 8 and 9 from the above worksheet.

  • For Wednesday, January 18, 2017, at the beginning of class: Here is another worksheet on Elementary Number Theory. For Wednesday's class, prove Propositions 10, 11, 12, 13 and 16.

  • For Monday, January 23, 2017, at the beginning of class: Do this homework. This homework will be graded (given a score out of 25), and the points will count towards your homework score.
    [Other homework which is not graded but just read and commented on by me counts as well, but only for having done it or not.]

    HERE are some worked solutions for this homework.

  • For Friday, January 27, 2017, at the beginning of class: In class on Monday 1/23, we worked on some propostions on this worksheet. For Friday, prove Propositions 22 and 23 and Lemma 31, and be ready to prove Theorems 29 and 30 in class.

  • For Monday, February 6, 2017, at the beginning of class: Do this homework. This homework will also be graded out of 30.

    HERE are worked solutions for this homework.

  • For Wednesday, February 8, 2017, at the beginning of class: Prove Theorem 30.

  • For Friday, February 10. On Monday, we worked on this worksheet. Prove Theorem 33 and Proposition 34.

  • For Monday, February 13. Practice negating mathematical expressions by negating the expressions in this worksheet.

  • For Wednesday, February 22. In class on Monday, we started working on the material on this worksheet. In class on Wednesday, you will be proving Proposition 2,3 and 4 in groups, so please come to class ready to write these proofs.

  • For Monday, February 27. Prove Propositions 3, 5 and 6 from the first worksheet on Fields. We will also be talking about how to prove that various things are (or aren't) fields, based on this worksheet. Read, practice and prepare for this.

  • For Friday, March 3. Let n be a natural number greater than 1 and let Zn be the set of equivalence classes of integers modulo n. Prove that Zn satisfies all of the axioms of being a field, except possibly for M3. Show that M3 holds for n = 2 and n = 3, but not for n = 4.

    (Note: We will keep the notation of Fn (in some font) for the case where (M3) holds.)

  • By request, here is a worksheet on Equivalence relations. You should already know (or already have done) most of things on this worksheet, up to the part which is the HW for 3/3 and the construction of the rational numbers.

  • For Wednesday, March 8, at the beginning of class. On Monday 3/6, we started working on this worksheet. Prove Corollary 40 from this worksheet (which should mostly involve remembering what was happening on the ENT Worksheet IV).

  • For Monday, March 13, at the beginning of class. Do this homework. It will be graded out of 25, according to the scheme indicated.

    HERE are worked solutions for this homework.

  • For Wednesday, March 15, at the beginning of class. Prove Propostion 43 from Elementary Number Theory, V. Also do Example 44. For Example 44, you must use the method discussed in class, using Proposition 43. You do not need to find x and y so that ax + by = d (where d is the gcd of a,b).

  • In class on Monday, March 13, We began to discuss this worksheet. Come to class on Wednesday ready to talk about this.

  • In class on Monday, April 3, We began to discuss this worksheet on induction.

  • Here is another worksheet with some induction problems, which should be easier problems than the last three on Induction Worksheet I.

  • For Friday, April 14, at the beginning of class, prove Propositions 11, 12, 15 and 16 from the second worksheet on Induction.

  • Here and here are worksheets on functions. We will spend the rest of the semester working on this material, and how much we get through will be determined.

  • For Friday, April 28, at the beginning of class, do this homework. This will be graded out of 25.

    Here are some worked solutions for this homework.