MATH 0413 Fall 2011

Schedule and Contact Information
Classes: M W F 11:00 - 11:50, Thackeray 627
Instructor: Alexander Borisov
Office: Thackeray 414
Email: borisov"at"pitt"dot"edu
Recitations: Tu Th 11:00 - 11:50, Thackeray 525
Recitation Instructor: Aziz Takhirov

Announcements and Homework
Midterm date is set: Friday, Oct. 21. Please refer to the Study Guide for more information.
Please refer to the List of Mini-quizzes for the most updated list of definitions and statements of theorems that can be asked on a mini-quiz.
Please refer to the List of Homework Assignments for the written homework assignments and due dates.

Prerequisites
Calculus II or equivalent, and some natural curiosity.

Text
We are using the free online book by Jiri Lebl, Basic Analysis, with Pitt supplements by Frank Beatrous and Yibiao Pan. Please let me know if you are interested in purchasing a hard copy for $16-$18.

Overview
The course covers the foundations of theoretical mathematics and analysis. Topics include sets, functions, number systems, order completeness of the real numbers and its consequences, and convergence of sequences and series of real numbers.

Core topics

  • Logic, proofs and quantifiers. Basic set theory. Functions and relations.
  • Elementary properties of the natural numbers; mathematical induction.
  • Axiomatic introduction to the ordered fields of rational and real numbers.
  • Elementary inequalities.
  • The Completeness Axiom; Archimedean Property of the real numbers; density of the rational and irrational numbers in the real numbers.
  • Countability of the rationals; decimal expansions of real numbers; uncountability of the real numbers.
  • Sequences and an introduction to series; the geometric series; limits; Limit Laws.
  • The Monotone Convergence Theorem.
  • The Bolzano-Weierstrass Theorem.
  • Cauchy sequences; Cauchy completeness of the real numbers.
  • Course Goals
    This course is a prerequisite for MATH 0420, and together they provide a rigorous foundation of the one-variable Calculus. Besides learning the topics listed above, you will get a better understanding of what constitutes a rigorous mathematical proof. This is a writing-intensive course, and during the semester you will be writing mathematical proofs of increasing degree of complexity. You will also learn to recognize and correct some mistakes in mathematical proofs.

    Notes on the Structure of the Course
    1. Recitations are an indispenable part of the course. Attendance and appropriate participation are required, part of the course grade comes from the recitation grade, which will be assigned by your recitation instructor.
    2. In the beginning of every lecture you will be asked to answer 2-5 simple questions on the definitions and statements of theorems. The primary purpose of these mini-quizzes is to ensure that you can follow the topic of each lecture.
    3. In addition to the daily mini-quizzes, you will have regular quizzes (during recitations, announced in advance), and homework. Their purpose is to help you achieve the mastery of the material required for solving problems on the midterm and the final, and to improve your mathematical writing skills. Some of these quizzes and homework may specifically address mathematical writing and proof comprehension/analysis.

    Grading
    Daily Mini-quizzes20%
    Regular Quizzes and Homework20%
    Recitation Grade10%
    Midterm (Date to be announced)20%
    Final (Date and place to be announced)  30%

    Academic Integrity
    Cheating/plagiarism will not be tolerated. Violations of the School of Arts and Sciences Policy on Academic Integrity will result in a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction.

    Homework Policy
    You may work with other students and/or use library/web resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.

    Disability Resource Services
    If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and the Office of Disability Resources and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.

    Helpful Links
    The following link has an excellent collection of interesting problems and well-written solutions, on relatively elementary topics. USA Mathematical Talent Search (USAMTS) Problems and Solutions