Homework assignments for Math 313-C1

Tenth homework assignment, due 12/01/2000:

• Section 10.5, problems 5, 10, 12, 16
• Additional problem for graduate credit: none

Ninth homework assignment, due 11/17/2000:

• Section 7.8, problem 39
• Section 8.5, problems 7, 8, 9
• Additional problem for graduate credit: none

Eighth homework assignment, due 11/10/2000:

• Section 7.8, problems 7, 12, 22, 23 (a)+(b)
• Additional problem for graduate credit: Section 7.8, 27

No homework due on 11/03/2000.

No homework due on 10/27/2000.

Seventh homework assignment, due 10/20/2000:

• Section 6.6, problems 10, 14, 19, 20
• Additional problem for graduate credit: Section 6.6, problem 18

Sixth homework assignment, due 10/13/2000:

• Section 6.6, problems 2, 3, 8
• Additional problem for graduate credit: Section 6.6, problem 15

Fifth homework assignment, due 10/06/2000:

• Section 5.8, problems 10, 11, 24
• Additional problem for graduate credit: none

Fourth homework assignment, due 09/22/2000:

• Section 3.6, problems 16, 19, 21
• Find out what's wrong with the following argument:

  Claim: If M is a finite set of people, then all the people in M are the same age.

  Proof: Let n be the number of people in M. We proceed by induction on n.

  1. n=1: There's nothing to show in this case.
  2. Assume that the claim is true for some n. Consider the set M with elements x_1,...,x_(n+1). Let A={x_1,...,x_n}, and let B={x_2,...,x_(n+1)}. By induction, all people in A are the same age, and all people in B are the same age. Hence, if y is an element of the intersection of A and B, then x_1 and y are the same age, and x_(n+1) and y are the same age. Hence, x_1 and x_(n+1) are the same age, and we are done.
• Additional problem for graduate credit: Prove that

  .

Third homework assignment, due 09/15/2000:

• Section 3.6, problems 5 (a) and (b), 6, 10, 13
• Additional problem for graduate credit:

  Fact: Let N and p be positive integers. Then there exists some positive integer M such that for every map f: {1,...,M} --> {1,...,N}, there exist positive integers n and q such that f(n)=f(n+q)=...=f(n+(p-1)q).

  Use this fact to prove the following claim: Let N, p, and q0 be positive integers. Then there exists some positive integer M such that for every map f: {1,...,M} --> {1,...,N} there exist positive integers n and q>=q0 such that f(n)=f(n+q)=...=f(n+(p-1)q).

Second homework assignment, due 09/11/2000 (note the new due date!):

• Section 2.4, problems 5, 11, 12 (assume that n is distinct from m), 17
• Additional problem for graduate credit: Section 2.4, Problem 13

First homework assignment, due 09/01/2000:

• Section 1.8: Problems 3, 4(a), 8
• Prove that a 2^n by 2^n board with one corner square removed admits a perfect cover of L-shaped tiles (see figures).

  2^n by 2^n board with one corner square removed
  Tile (consists of three small squares)

• Additional problem for graduate credit: Section 1.8, Problem 4(b)