CRN 11244

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"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."
Lewis Carroll

  • Time and Place. TR 13:30-14:50 in LART 108
  • Instructor. Helmut Knaust, Bell Hall 219,, 747-7002
  • Prerequisites. The course requires a certain level of mathematical maturity that you should have gained by, for instance, having thoroughly and successfully grappled with the concept of infinity in your Calculus II course (which is the formal prerequisite for this course).
  • Course Objectives. This is a Foundations course. This means that hardly any prior knowledge is required. The class prepares you "do" mathematics on your own and enables you take more advanced classes or read rigorous mathematical textbooks. You should expect (and I will expect) that you make considerable progress in the following areas:
  1. Make sense of an abstract definition by analyzing it carefully and constructing examples.
  2. Make sense of a mathematical statement and be able to bring to bear a variety of strategies for constructing its proof.
  3. Be able to recognize a rigorous proof when you read one. Conversely, be able to pick out the weak spot(s) in a less rigorous argument. Be able to fill in details in a sketchy proof.
  4. Once you have devised a proof, be able to write it down in a clear, concise manner using correct English and mathematical grammar.
  5. Be able to present and defend a proof to a group of your peers.
  • In-class Activities and Presentations: Mathematics is not a spectator sport. Therefore I do not give lectures. I will call on students to give presentations of exercises in the class notes. I will also regularly ask for volunteers to present solutions to problems at the blackboard. Your presentations are the most important part of the course. Your chances of passing the course without spending a significant amount of time on preparing problems to present in class are zero: Your in-class work, evaluated for both quality and quantity, will account for 30% of your grade.
  • Tests. Exams will be given on the following days: Thursday, September 29, and Tuesday, November 15. Each exam counts 20% of your grade. You may not leave the classroom during tests or the final.
  • Final Examination. The final exam on Thursday, December 8, at 13:00-15:45, is comprehensive and mandatory. It counts 30% of your grade.
  • Time Requirement: I expect that you spend an absolute minimum of at least six hours outside of class on reviewing your own notes and preparing problems for the next class. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
  • Attendance: Due to the course structure, attendance is mandatory. An unexcused absence will result in an exercise/task grade of 0 for the day of the absence. Six absences (excused or unexcused) will lead to dismissal from the class with a grade of "F".
  • Information on the Web: This syllabus and ancillary material can be found on my web page
  • Ground Rules:
    • The notes distributed in class contain “exercises" and “problems". Students will solve these problems at home and then present the solutions in class. The instructor will call on students at random to present “exercises"; he will call on volunteers to present solutions to the “problems".
    • When in the audience, students are expected to be actively engaged in the presentation. This means checking to see if every step of the presentation is clear and convincing, and speaking up when it is not. When there are gaps in the reasoning, the students in class will work together to fill the gaps.
    • The instructor serves as a moderator. His major contribution in class will consist of asking guiding and probing questions. He will also occasionally give short presentations to put topics into a wider context, or to briefly talk about additional concepts not dealt with in the notes.
    • Students may use only the class notes and their own notes taken during the semester; they are not allowed to consult other books or materials. Students must not talk about assignments to anyone other than class participants and the instructor. Students are encouraged to collaborate with other class participants; if they do, they must acknowledge other students’ contributions during their presentation. Exemptions from these restrictions require prior approval by the instructor.
    • The instructor is an important resource. He expects frequent visits from all students in class during his office hours – many more visits than in a “normal" class. Among other things, students probably will want to come to the instructor’s office to ask questions about concepts and assigned problems, they will probably occasionally want to show the instructor their work before presenting it in class, and they probably will have times when they just want to talk about the frustrations they may experience.
    • It is of paramount importance that we all agree to create a class atmosphere that is supportive and non-threatening to all participants. Disparaging remarks will be tolerated neither from students nor from the instructor.
  • Drop Policy. The class schedule lists Friday, October 28, as the last day to drop with an automatic "W". After the deadline, I can only drop you from the course with a grade of "F".
  • Graduate Credit: If you take this undergraduate course for graduate credit, you will additionally write up solutions to some extra problems. This assignment is due on December 1, and will be graded for an additional 20% of your grade. (Since your possible score therefore increases to 120%, I will divide your total score by a factor of 1.2 to get to a 100% scale.)
  • Academic Integrity. All students must abide by UTEP's academic integrity policies. For detailed information visit the Office of Student Conduct and Conflict Resolution (OSCCR) website. Academic Integrity is a commitment to fundamental values. From these values flow principles of behavior that enable academic communities to translate ideals into action.” Specifically, these values are defined as follows:
    • Honesty: advances the quest for truth and knowledge by requiring intellectual and personal honesty in learning, teaching, research, and service.
    • Trust: fosters a climate of mutual trust, encourages the free exchange of ideas, and enables all to reach their highest potential.
    • Fairness: establishes clear standards, practices, and procedures and expects fairness in the interaction of students, faculty, and administrators.
    • Respect: recognizes the participatory nature of the learning process and honors and respects a wide range of opinions and ideas.
    • Responsibility: upholds personal responsibility and depends upon action in the face of wrongdoing.
  • Military Service. If you are a military student with the potential of being called to military service and/or training during the course of the semester, you are encouraged to contact the instructor as soon as possible.
  • Counseling Center. You are encouraged to go to Counseling and Psychological Services (202 Union West) for personal assistance as you work through personal concerns. Confidential counseling services are offered in English or in Spanish.
  • Disabilities. If you have a disability and need special accommodation, please contact the Center for Accommodations and Support Services (CASS). The Center aspires to provide students accommodations and support services to help them pursue their academic, graduation, and career goals. Phone 747-948. E-mail:

Class Notes

Oct. 20 | Oct. 13 | Oct. 11 | Oct. 6 | Sept. 27 | Sept. 15 | Sept. 13


  • Also: Supplementary problems posted here. Section 3 added!
    Done: S 1.1; Section 2

  • 11/22: 4.47, 4.58, 5.29, 5.33, 5.36 - 5.66
  • 10/25: 4.47, 4.58. 5.3., 5.11 - 5.13, 5.19 - 5.29
  • 10/11: 4.45-4.59
  • 10/6: 3.11; 4.39-4.59
  • 10/4: 3.11,3.16; 4.30,4.31,4.36,4.39-4.59
  • 9/27: 3.11,3.16; 4.29-4.52
  • 9/22: 3.11,3.16; 4.20-4.32
  • 9/20: 3.11,3.16; 4.1-4.32
  • 9/15: 3.5,3.8,3.10,3.11,3.14,3.16; 4.1-4.18
  • 9/13: 3.5-3.16
  • 9/6: 3.1-3.16
  • 9/1: 2.3-2.8; 3.1-3.8
  • 8/30: 1.19-1.27; 2.1-2.8
  • 8/25: 1.12.2-1.27
  • 8/23: 1.1-1.15


Polya's Horses | Supplement | Polya's How to Solve It | Checking proofs, and Greek letters | How to Write Mathematics, by Martin Erickson | Ivan Niven, A simple proof that π is irrational. Bull AMS 53 (1947), 509.

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