Class

• Mon. 1:30-2:50
• Wed. 1:30-2:50
• Hegeman 201

Instructor

• Ethan Bloch
• bloch 'at' bard 'dot' edu
• Albee 317
• 845-758-7266

Office Hours

• Mon. 10:30-11:30 & 2:30-4:00
• Tue. 11:00-12:30
• Wed. 2:30-4:00

Text

• Bloch, Ethan, 'Proofs and Fundamentals: A First Course in Abstract Mathematics,' 2nd ed., Springer, 2010

Website

• http://math.bard.edu/bloch/math261A.shtml (includes updated list of assignments)

Errata for Textbook

Communications

• Urgent announcements may be sent out via campus email, so make sure you either check your Bard email regularly or have your Bard email forwarded to the email address of your choice.

Attendance

• It is expected that students attend all classes. Bring the text to each class.

Homework

• Homework will be assigned at the end of every class. Turn in the homework at the start of the next class. Late assignments will not be accepted, except in genuine emergency situations.

• Prior to spring break, homework may be handed in handwritten. After spring break, homework should be typed in (mathrm LaTeX), using the Bard Macros style file. (mathrm LaTeX) will be discussed in class. If you need assistance with (mathrm LaTeX), please ask the instructor.

• You are encouraged to work with other students in solving the homework problems. However, for the sake of better learning, as well as honesty, please adhere to the following guidelines:
  
• Write up your solutions yourself.
• Acknowledge in writing anyone with whom you work and any assistance you receive.
• Acknowledge in writing any revisions of your work based upon solutions given in class.
• Do not make use of any solutions found on the web.

• Failure to indicate collaboration, assistance or sources will be construed as plagiarism.

• The use of homework solutions found on the web or elsewhere will be treated similarly to plagiarism on exams.

• Your solutions should be written clearly and carefully, as described below.

Exams

• Each exam will have an in-class, closed-book part, and a take-home, open-book part.

• Midterm Exam (in class): Tue., Mar. 15
• Midterm Exam (due): Thur., Mar. 17

• Final Exam (in class): Thur., May 19
• Final Exam (due): Tue., May 24

• This course is an introduction to the methodology of mathematical proof, and to fundamental topics that are used throughout mathematics. Topics for writing proofs include the logic of compound and quantified statements, direct proof, proof by contradiction and mathematical induction. Fundamental mathematical topics include basic set theory, functions, relations and cardinality.

• The minimal prerequisite is a mathematics course at the level of Calculus II (Math 142); a mathematics course at the level of Linear Algebra and Ordinary Differential Equations (Math 213) is preferred.

• If you are unsure whether this course is an appropriate course for you, please consult with the instructor.

• If you have any problems with the course, or any questions about the material, the assignments, the exams or anything else, please see the instructor about it as soon as possible. If you cannot make any of the scheduled office hours, please make an appointment for some other time. To make an appointment, or to discuss anything, talk to the instructor after class, or send him an email message, or just stop by his office.

• Calculators and computers are not needed will not be needed during class.

• Use of a computer will be needed for typing the homework in (mathrm LaTeX), which will be required after spring break, and will be optional before spring break. (mathrm LaTeX) will be discussed in class.

• Electronic devices, including cell phones, tablets and laptop computers, may not be used during class, other than to read the text.

• The Mathematics Study Room is open Sunday-Thursday, 7:00pm-10:00pm, in RKC 111.

• The Mathematics Study Room is staffed by undergraduate mathematics majors who are available to answer your questions. You can go to the study room to work on your homework, and then ask for help as needed.

• If you need additional help beyond office hours and the Mathematics Study Room, you can meet with a tutor. This service is provided by the Bard Learning Commons.

• If you want to meet with the tutor assigned to this course, you can either go to the tutor's office hours, or email the tutor to set up an individual appointment, or both.
• Tutor: Andres Mejia
  Office hours: Wednesday, 6:00-8:00 pm, Mathematics Common Room (third floor of Albee).
  Email: am8248 'at' bard 'dot' edu.

• Grades will be determined roughly 50% by the homework assignments and 50% by the exams. Class participation will be taken into account positively, especially in cases of borderline grades.

• Grades will be determined by work completed during the semester, except in cases of medical or personal emergency. There will be no opportunity to do extra credit work after the semester ends.

• This course is graded using letter grades. If you want to take the course Pass/Fail, you must submit a request to do so to the RegistrarÕs Office during the Add/Drop period.

• Students with documented learning and/or other disabilities are entitled to receive reasonable classroom and testing accommodations. If you need accommodations, please adhere to the following guidelines:
  
• Discuss your needs with the instructor at the beginning of the semester.
• Provide documentation as appropriate.
• Contact the instructor at least one week prior to each quiz, exam or instance of accommodation.

• If you need to miss a class for any reason (sports team, religious holiday, etc.), it is your responsibility to contact the instructor and find out about the material and assignments you missed.

• Wed., Feb. 11: End of Drop/Add period.

• Mon., Mar. 16 - Fri., Mar. 20: Spring break.

• Mon., Apr. 27 & Tue., Apr. 28: Advising days (no classes).

• Tue., Apr. 28: Last day to withdraw from a class.

• Wed., Apr. 29: Senior projects due.

• Tue., May 19: Last day of classes.

• Everyone makes honest mathematical mistakes, but there is no reason to get in your own way by writing your proofs with incomplete sentences and other grammatical mistakes, by using undefined symbols for variables or by engaging in other forms of sloppy writing. Mathematics must be written carefully, and with proper grammar, no differently from any other writing.

• This course will offer many opportunities to practice the careful writing of mathematical proofs. Properly written proofs require the writer to observe the following basic points.
  
• Write your homework assignments neatly and clearly.
• Justify each step in a proof, citing the appropriate results from the text as needed.
• Use definitions precisely as stated.
• Use correct grammar, including full sentences and proper punctuation.
• Be very careful with quantifiers.
• Strategize the outline of a proof before working out the details; the outline of a proof is always determined by what is being proved, not by what is known.
• Distinguish between scratch work and the actual proof; scratch work can be in any order, but the actual proof always starts with what is known and deduces the desired result.
• Proofs should stand on their own; check your proofs by reading them as if they were written by someone else.

• Proofs-based mathematics courses are very different from computation-based mathematics courses such as Calculus. The ways you studied, did homework and took exams in computation-based courses was appropriate for those courses, but not for proofs-based courses. Approach proofs-based courses with the idea that you will be doing things differently from what you did in computation-based mathematics courses.

• The material in this course is much more abstract, and requires much more precision in both studying and problem solving, than the material you saw in courses such as Calculus. Some students find the material in proofs-based courses more difficult than the material in Calculus courses, and, for some students, a proofs-based course such as this one constitutes the first time that they found a mathematics courses challenging, which can be intimidating at first, but is in fact completely normal. Everyone, including the very best mathematicians, reaches a level of mathematics that he or she finds difficult; what varies from person to person is only what that level is. If you made it this far in mathematics and you only now first encounter substantial difficulty in learning the material, you are doing fine.

• In general, the more advanced you get in mathematics (or any subject), the larger the percentage of learning that takes place outside of class, including from the textbook, from other sources, from office hours, from tutors and from your fellow students (not necessarily in that order).

• In Calculus courses, where the material can mostly be learned in class, reading the textbook is not necessarily very important. By contrast, in proofs-based courses reading the textbook carefully, and seeking help with those parts of the textbooks that you find difficult, is crucial.

• In Calculus courses, solutions to homework exercises are usually written as a collections of equations, with little or no words explaining the solution. By contrast, rigorous proofs are, fundamentally, convincing arguments, and to make a good argument, words are needed to direct the logical flow of the ideas; to explain what is assumed and what is to be proved; and to state what previous results are used. In particular, rigorous proofs are written using full sentences, and with correct grammar and punctuation, because doing so helps make the arguments more clear and precise.

• In Calculus courses, solutions to homework problems are usually written directly, with little revision. By contrast, rigorous proofs should be written the same way a paper in a humanities course is written, by first making an outline (often called 'strategizing a proof'); then sketching out a rough draft; then revising the draft repeatedly until the proof works; and, lastly, writing the final draft carefully, and, when required, typing it (via (mathrm LaTeX)).

• Revising a draft of a rigorous proof should be done exactly as revising a draft of a paper in a humanities course, which is to read it as if you are not the author, but rather as if you are someone else in the class, and making sure that each sentence makes sense as written, without recourse to unwritten explanations.

• Learning to write rigorous proofs takes time, and you should not expect to master it instantly.

• A very good way to improve your skill at writing proofs, and also to do better on the homework in this course, is to bring a draft of each homework assignment to office hours before you write up and submit the final draft.

• This style file, called bardmacros.sty, is designed for homework assignments in upper level mathematics classes, such as Proofs and Fundamentals, Abstract Algebra, Real Analysis and Topology. It takes care of a number of formatting issues such as exercises, definitions and the like, and has specific macros for various upper level classes. There is a template, a brief manual for this style file and writing guidelines for proofs-based homework. Click below to download these files.

• You can download all four files in one .zip file (recommended), or you can download each file individually (some as .zip files and some as .pdf files). The .zip files need to be uncompressed prior to use.

• There are sometimes problems downloading these files with the Chrome web browser. If you have problems with Chrome, try a different web browser.

• All four files together:
  

• Individual files:
  
• Writing guidelines for homework with proofs
  

• This video shows how to use the Bard macros style file.