Last updated May 4, 2008

 

MATH 304, Prove it

Spring 2008

     
         
       

Class Schedule

TR 10:10-11:25, 224 Malott Hall

Lecturer

Robert Connelly, 433 Malott Hall, (607) 255-4301, connelly@math.cornell.edu

Office hours: Monday 3-4, Friday 2-3;  433 Malott Hall

Teaching Assistant

Michael O'Connor  

Office hours: Wednesday 2-4; 218 Malott Hall

Text

 How to Read and Do Proofs by Daniel Solow (recommended).

Some course notes and definitions.

  • Glossary
  • Symbolic Logic: Propositional Calculus
  • Symbolic Logic: Predicate Calculus
  • A quick discussion of equivalence relations from Halmos's "Naive Set Theory", which is on reserve in the Math Library.
  • Here is some advice by George Polya on how to solve problems and prove things.
  • Here is a good discussion of infinite cardinals in Halmos's book above.  Two copies of this book are on reserve in the Mathematics Library.
  • A CD for the book "An interactive introduction to Mathematical Analysis" by Jonathan Lewin is on reserve in the Mathematics Library.  Part I gives good advice for learning how to do proofs.  We will have some exercises from this book.
  • Here is more information about the creation/definition of the natural numbers as discussed in class.  
  • Here is a discussion of the creation/definition of the integers.
  • Here is a discussion of the creation/definition of the rationals.
  • Here is a discussion of the creation/definition of the real numbers.
  • Here are things about cardinal numbers.
  • Here are some definitions and theorems about cardinality to help with the homework.
  • Here are some problems for practice for the Prelim on March 13.  
  • Here are solutions to the midterm Prelim.  The median score was 69 out of a possible 100.
  • Here is a collection of sample questions for the Final exam.  Enjoy.

Course Information

We will have regular weekly homework due every Tuesday, at least one in-class Prelim, and a Final Exam.

We had an in-class preliminary exam on logic, cardinal numbers, and induction on Thursday, March 13.

The Final Exam will be Tuesday, 13-MAY, from 7:00 PM to 9:30 PM in Rockefeller 104.  It will cover all the material in the course with a bit more emphasis on the material since the mid-term prelim.

The last class on May 1 will be a review.  Please have questions ready.

Homework Assignments

  • Read Chapter 1 in Solow, and do Exercises 1.1, 1.2, 1.4, 1.8.  Prove that any projective plane has at least 7 points.  (Use no more than one page, typewritten, clearly and succinctly presented.)  Find the simplest proof of Pythagoras's Theorem you can find.  Due Tuesday, January 29, 2008, in class. Solutions.
  • Read Chapter 2 in Solow, and Halmos's chapter on equivalence relations.  These exercises are due February 5, 2008 in class.  This involves a discussion of equivalence relations. Solutions.
  • Read Chapter 2 and Chapter 3 in Lewin's book.  Do exercises 3, Section 2.1.5, exercise 3 in Section 2.3.8, exercise 2 in 3.6.6, exercise 4 in Section 3.7.3, exercise 7 in Section 3.8.4, exercise 9 in Section 4.3.15. Due February 12 in class. Solutions.
  • Read Chapters on the Schröder-Bernstein Theorem and countable sets in Halmos's book.  (CORRECTION) These exercises are due February 26, in class. THE EXERCISES HERE ARE CORRECTED from the earlier version. Solutions.  Note the delay.  We will have a short quiz (not graded but collected) in class on Thursday, February 22.  It will cover cardinal numbers.
  • We will have a short quiz (not graded but collected) in class on cardinal numbers on February 28.
  • Read Chapter 11 in Solow's book on mathematical induction.  Here are some problems on our discussion of the Schröder-Bernstein Theorem and cardinal numbers as well as induction.  Due in class in class Thursday, March 6. Solutions.
  • We will have a short quiz that will be graded in the last 20 minutes of class covering mathematical induction on Thursday, March 6.
  • Read Appendix C.1 in Solow and do exercises C.2, C.4, and C.5. Which positive  multiples of 5 pounds are NOT possible with garbage tags of denominations of 20 pounds and 35 pounds? (Hint:  If you know that enough cosecutive multiples of 5 ARE positive combinations of 20 and 35, then all further multiples are positive combinations of 20 and 35.  More hints in class.  This problem can be handed in on Thursday.)  Due April 1 (no fooling).Solutions
  • There will be a short quiz that will be graded in the last 20 minutes of class covering number theory and a little bit  of Appendix D.1 on Thursday, April 3.
  • Read Appendix D in Solow and do exercises D.3, D.4.   Due April 8. (D.9, D11, D12 are due next week.) Solutions.
  • There will be a short quiz that will be graded in the last 20 minutes of class covering Appendix D.1 and D.2 and glb's and lub's on Thursday, April 10.
  • Do Exercises D.8, D.9, D.11, D.12 in Solow.  Due April 15.Solutions
  • Do the three problems here.  Due April 22. Solutions. There will be a short quiz covering open sets and continuity that will be graded in the last 20 minutes of class on Tuesday, April 22.
  • Do the Exercises here.  Due April 29 in class.  You can read about the fractal dimension of sets with lots of examples here.Solutions