Comps paper summary/introduction: due on the last class day.
Day 12 (4/23/12): LaTeX test. (See day 11 for sample test.)
Day 11 (4/16/12): More LaTeX practice. The test will be on Day 12. Here's a sample test. The actual test will be different from but very similar to the sample test. You will be allowed to use an online LaTeX reference during the test.
Day 10 (4/9/12): Download the file Practice2.tex at the bottom of this page and typeset it with any LaTeX software.
Comps exam: Wednesday, April 4, 2012, 5-8 pm, Fowler 309. You may be able to start a bit earlier or later if you ask me in advance.
Day 9 (4/2/12): Junior Writing Requirement test #3. Topic: Give the standard definition of linear independence. Give examples of a linearly dependent, and a linearly independent, set of vectors, with justification. Then prove that a set of vectors is linearly independent if and only if none of the vectors is a linear combination of the others. Also explain, with justification, whether in a linearly dependent set of vectors, every one of the vectors is a linear combination of the others. Pretend you are writing a short book section or paper: give a clear and mathematically precise exposition.
Day 8 (3/26/12): Comps2008.doc. Comps2009.doc Add your names to the sign-up sheet above.
We will also work on LaTeX. Download the file Practice1.tex at the bottom of this page and typeset it with any LaTeX software.
Some LaTeX references:
LaTeX Symbols (large file: 4MB) (copied from: http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf)
Quoted from http://www.latex-project.org/intro.html : LaTeX is based on the idea that it is better to leave document design to document designers, and to let authors get on with writing documents.
Day 7 (3/19/12):
Comps2007.pdf . Put your name and the problems you will do on the sign-up sheet above.
Start working on the "comps paper assignment": Pick sources (journal, book, etc.) for your comps paper, and then write a 1-2 page description, in LaTeX, of what your paper will be about. Due date: Day 12.
Day 6 (3/5/12): Test #2 for the Junior Writing Requirement (for topic, see below).
Comps2007.pdf . Put your name and the problems you will do on the sign-up sheet above.
Day 5 (2/27/12):
Comps2006.pdf . Put your name and the problems you will do on the sign-up sheet above.
Test #2 for the Junior Writing Requirement will be on Day 6. Topic:
Define what it means for two sets to have the same cardinality. Give a simple example -- with proof -- of a set that has the same cardinality as a proper subset of itself. Then briefly give some motivation for your definition and explain why it "makes sense" even though your example above is counter-intuitive. Define countable and uncountable. Then prove one of the following: (i) the set of integers is countable; or (ii) the set of real numbers is uncountable. In your proofs, if you construct a function that you claim is a bijection, you should prove that it's a bijection. You may assume the reader already knows the definitions of "bijection", "one-to-one", and "onto".
Day 4 (2/13/12):
Comps2006.pdf . Put your name and the problems you will do on the sign-up sheet above.
In-class test on the Junior Writing Requirement (for topic, see Day 2 below).
Day 3 (2/6/12):
Comps2005.pdf . Put your name and the problems you will do on the sign-up sheet above. (Sign up for the ones you find difficult.) At the bottom of this page there are links to lists of topics covered in each of the five topics on the comps exam.
Solutions to the 2004, 2005, and 2008 exams (apparently, some or all of them were written by students; and not every problem is there)
Day 2 (1/30/12):
Comps2004.pdf . We will spend time reviewing for the comps exam by you writing up your solutions ahead of time and then copying them on the board in class. Put your name and the problems you will do on the sign-up sheet above. Do as many problems as you wish; try to do ones you find difficult, not the easy ones!
Our first Junior Writing test is listed below. The test will be given on "Day 4" (2/13/11).
Give the definition of each of the following: (i) a ≡ b (mod n); (ii) equivalence relation; (iii) equivalence class. For each definition give an example that illustrates and "sheds light on" the definition. Prove that "congruence mod 2" is an equivalence relation (make sure your proof is consistent with your definitions). List all equivalence classes for the equivalence relation "congruence mod 2"; explain and justify how you know you've listed all the equivalence classes for this relation. Pretend you're writing a short section of a discrete mathematics book; give a clear and well organized exposition, using precise mathematical language.
Day 1 (1/23/12):
Practice test for writing: Give the definitions of "odd", "even", and what it means for two integers to have the "same parity". Give a short example to illustrate each of your definitions. Then prove that two integers have the same parity iff their sum is even. Pretend you are writing a very short section of a book for a Discrete Mathematics course; use clear, but precise and formal language. (I will grade but not count this test.)
Lists of topics covered in each of the five sections on the comps exam: