HOMEWORK FOR SET THEORY STUDENTS

Math 4730: Set Theory, Spring 2009

Homework

Assignment	Assigned	Due	Problems
HW1	1/14/09	1/23/09	Read pages 1-11. 1. Do Exercise 1.3.1 2. Do Exercise 1.3.3 (b) 2. Do Exercise 1.3.6
HW2	1/21/09	1/28/09	Read pages 12-27. For practice, try problems 1.4.3, 2.1.1, 2.2.4, 2.2.7. 1. 2.2.1 2. 2.2.2 3. 2.2.8(a) and first claim of 2.2.8(b)
HW3	1/29/09	2/4/09	Read pages 28-32. For practice, try problems 2.3.1, 2.3.10 (first part), 2.3.12, 2.4.1 (all). 1. 2.3.5 2. 2.3.6(a) 3. 2.4.1(b)(d)
HW4	2/5/09	2/11/09	Read pages 33-37. For practice, try problems 5.3, 5.4, 5.7. 1. (a) Let F(x,y)=x+y and G(x,y) = xy be functions from R^2 (the real plane) to R (the real line). Draw pictures of some of the equivalence classes of the kernels of these functions. (b) Now let H(x,y) = (F(x,y),G(x,y)) be the function from R^2 to R^2 with coordinate functions F and G. Show that the kernel classes of H have size at most 2. 2. How many orderings on {0,1,2} are there? 3. What are the least and greatest number of pairs that can occur in an ordering of an n-element set? (Problem to think about: is every intermediate value equal to the number of pairs of some ordering of an n-element set?)
HW5	2/12/09	2/18/09	Read pages 39-54. For practice, try problems 3.1.1, 3.2.2, 3.2.3, 3.2.4. In all HW problems, "n+1" is supposed to be "S(n)". (At this point of the book, addition has not been introduced.) 1. Do 3.2.1 2. Do 3.2.6 3. Do 3.2.8
HW6	2/18/09	2/25/09	Read pages 65-73. 1. Prove that m^n+k = mⁿ m^k. (You may need to prove some lemmas first.) 2. Show that the function F:NxN --> N* defined by F(m,n) = 2^m(2n+1) - 1* is a bijection. Conclude that \|NxN\| = \|N\|. 3. Show that the real line has the same cardinality as the real plane. (Hint: By the Cantor-Bernstein Theorem you only need to find 1-1 functions in each direction. For a 1-1 function from the plane to the line, try mapping a point (x,y) in the plane to the real number obtained by interlacing the digits of x and y. Be careful to explain exactly what you mean, noting that some real numbers have more than one decimal representation.)
HW7	2/25/09	3/4/09	Read pages 74-78 and 90-92. For practice, show that is S is an infinite subset of N, then the power set P(S) is uncountable. 1. Do 4.3.3. 2. Show that the set of all equivalence relations on N is uncountable. 3. Show that the set of all linear orders on N is uncountable.
No HW	3/4/09	3/11/09	Read pages 103-114. For practice, think about exercises 6.1.1-6.1.5 and 6.2.1-6.2.8.
HW8	3/12/09	3/18/09	Read pages 114-122. 1. (a) Give an example of a set that is transitive, but not well-ordered by epsilon. (b) Give an example of a set that is well-ordered by epsilon, but is not transitive. 2. Show that the lexicographic product of two well-ordered sets is well-ordered. (The lexicographic product is defined in Lemma 4.6 of Chapter 4.) 3. Prove the "if" part of Exercise 2.6 of Chapter 6. (Hint: Use an observation from the proof of Theorem 2.10 of Chapter 6.)
HW9	3/19/09	4/1/09	1. Do Exercise 6.5.3. 2. Do Exercise 6.5.6. 3. Do Exercise 6.5.9.
HW10	4/2/09	4/8/09	Read pages 124-127. 1. Do Exercise 6.5.8(a). (Use transfinite induction on gamma.) 2. Do Exercise 6.5.11(a)(d). 3. Do Exercise 6.5.15.
HW11	4/10/09	4/15/09	Read pages 129-136. 1. Do the part of Exercise 7.1.2 that concerns products of ordinals. 2. Do the part of Exercise 7.2.4 that concerns exponentiation of ordinals. 3. Do Exercise 7.2.5.
HW12	4/16/09	4/24/09	Read pages 137-143. 1. Do Exercise 8.1.9. 2. Do Exercise 8.1.12. (Use Zorn's Lemma.) 3. Do Exercise 8.1.16.