Sign-up sheet for old comps exam problems.
The LaTeX exam will be on our last class day, 4/29/13. Here's a LatexTest-sample.pdf ; the source file for it is at the bottom of the page. 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.
The "comps project description" paper is due Monday 5/6/13; it will need to be written up in LaTeX. I'll talk more about this in class.
Day 10, 4/22: Practice2.tex , Practice2.pdf (at the bottom of this page).
Homework due Monday 4/22 (no need to turn in anything): Download the file Practice1.tex (at the bottom of this page), typeset it in LaTeX, read it carefully, and then "play around" with LaTeX. Also look at the Latex references below; I recommend reading LaTeX Primer a bit (the more the better).
Day 10, 4/17 : First half hour: learn LaTeX. Then, 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 S of vectors is linearly independent if and only if no vector in S is a linear combination of the other vectors in S. 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 well-organized exposition using mathematically precise language.
Some LaTeX references:
LaTeX Symbols (4MB) (copied from: http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf)
http://www.texify.com/
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 9, 4/8/13: Comps exam: 5:10-8:10 pm, in Fowler 302.
Day 8, 4/1/13: Comps2009.pdf
Day 7, 3/25/13: Comps2008.pdf
Day 6, 3/18/13: Comps2006.pdf , Comps2007.pdf . Put your name and the problems you will do on the sign-up sheet above.
Day 5, 3/4/13:
Continue with Comps2005.pdf (half an hour)
Writing test #2: Define what it means for two sets to have the "same cardinality". Give an example, different from part (i) below, of a set that has the same cardinality as a proper subset of itself. Prove that the two sets in your example indeed have the same cardinality (just give an appropriate bijection, but no need to prove it is a bijection). Define "countable" and "uncountable", and give an example for each. Then prove one of the following: (i) the set of integers is countable; (ii) the set of real numbers is uncountable. In your proof, 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". Pretend you're writing a short article or book section; give a clear and well organized exposition, using precise mathematical language.
Day 4, 2/25/13: 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!)
Day 3, 2/11/13:
Continue with Comps2004.pdf (half an hour)
Junior Writing test #1 (one hour). (See Day 2 for description.)
Day 2, 2/4/13.
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're willing to do on the sign-up sheet above. Do as many problems as you can; try to do the ones you find difficult, not the easy ones. If someone else is planning to do a certain problem, it doesn't mean that you shouldn't do it. In class it's enough for each problem to be done by only one person; but you should do ALL problems at home. At the bottom of this page there is a list of topics that are covered on the comps exam. There is also a link to review notes that one of our math majors (Megan Upp) wrote up in 2011 or 2012.
Our first Junior Writing test will be on "Day 3" (2/11/13): 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 on the set of all integers (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 or a comps paper; give a clear and well organized exposition, using precise mathematical language. Do not write in "abbreviated form," like we often do on homework or on the board.
Day 1, 1/28/13: Practice test for junior writing requirement.
Give the definitions of "odd", "even", and what it means for two integers to have the "same parity". Give a brief 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, precise and formal language. Do not write in "abbreviated form," like we often do on homework or on the board. You may assume every integer is odd or even but not both. (I will grade but not count this test.)
Lists of topics covered in each of the five sections on the comps exam
Review notes by Megan Upp (I haven't checked these at all)