# Math 361 (Set Theory) - Fall 2007Homework Assignments

 Due Assignment 12/21 FINAL EXAM (8:00-11:00 AM in Beck 201) 12/21 If a linear transformation V→W is onto, show that V = ker(f) ⊕ W' and f:W'≅W. Hint: Find a basis of ker(f) and use Zorn's Lemma to construct a basis of V containing it. 12/10 (a) Show that every finite linearly ordered set (A,<) is well ordered; (b) Fix a set A, and let D be the set of ordinals isomorphic to a subset S of A (under some well-ordering of S). Show that D is an ordinal, and that D is the smallest ordinal not dominated by A. (This includes showing that card(D)>card(A).) 11/28 MIDTERM 11/26 (a) If X0∈X1∈…∈ Xn, show that Xn∉X0. (b) If Ω is an ordinal, show that every element β∈Ω is an ordinal. 11/19 (a) Show that N×N is well-ordered by lexicographic ordering. (b) If A is well-ordered and f:A→A is order-preserving, show that either some initial segment is not in f(A) or else f is the identity map. 11/14 (p.165, #34) Assume that 2<κ≤λ and λ is infinite. Show that 2λ=κλ 11/12 Use Zorn's Lemma to show that every vector space has a basis. 11/7 (p.161, #27) Determine the maximum cardinality of sets of disjoint disks, circles, and figure 8's (respectively) in the plane. 11/5 (p.144, #10-11) Prove the exponentiation rules κλμ=κμκλ and (κλ)μ=κμλμ. 10/31 Let A be a subset of a countable set B. (i) Show that A is countable; (ii) If A is not finite, show that card(A)=ℵ0. 10/29 If A is a finite set, show that every surjection A→A is a bijection. 10/17 MIDTERM and the following take-home problem: Given a Cauchy sequence {rn} in Q, show that (∃A∈R) rn → A. 10/15 Use Trichotomy for < on Z to prove that Trichotomy holds for < on Q. 10/10 Let f: X → Y be a function and R an equivalence relation on X. Suppose that if x'Rx" then f(x')=f(x"). Show that there is a unique function ƒ: X/R → Y such that if c=[x] then ƒ(c)=f(x). 10/8 Show that addition of natural numbers is associative. (Recall that A(n)=m+n is defined by recursion.) 10/3 If m,n are natural numbers, and m+=n+, show that m=n.   (Recall that A+=A ∪ {A}.) 10/1 (a) Show that X² (defined on p.52) is in 1-1 correspondence with X×X for all X. (b) If S:4→6 is Sn=n+1, describe ∏ Sn. (Recall that 4={0,1,2,3}.) 9/26 (a) If R⊆A×B is onto, use AC to prove that there is a function f:B→A such that R°f=IB. (b) (p.55 #31) Prove that the two forms of the Axiom of Choice are equivalent. 9/24 Let R be a relation. (a) Show that the domain of R-1 is range(R); (b) Show that (R-1)-1=R; (c) If R is a one-to-one function and x is in domain(R), show that R-1(R(x))=x. 9/17 Translate and prove: (∀S) if S ≠ ∅ then (∃! M)(∀x)(x∈M iff (∀A∈S) x∈A ). 9/12 Show that the operations +=XOR and ×=∩ make the power set ℘(X) into a commutative ring with 0=∅ and 1=X, and that a^2=a and a+a=0 for all a∈℘(X).

## Homework is due one week after being assigned

Charles Weibel / weibel@math.rutgers.edu / December 12, 2007