Homework Assignments

Due | Assignment | |

12/21 | FINAL EXAM
(8:00-11:00 AM in Beck 201)
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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.
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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).)
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11/28 | MIDTERM
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11/26 | (a) If X_{0}∈X_{1}∈…∈
X_{n}, show that X_{n}∉X_{0}. (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.
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11/14 | (p.165, #34) Assume that 2<κ≤λ and λ
is infinite. Show that 2^{λ}=κ^{λ}
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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
κ^{λμ}=κ^{μ}κ^{λ}
and
(κλ)^{μ}=κ^{μ}λ^{μ}.
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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)=ℵ_{0}.
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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 {r _{n}} in Q,
show that (∃A∈R) r_{n} → A.
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10/15 | Use Trichotomy for < on Z to prove that
Trichotomy holds for < on Q.
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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 S _{n}=n+1, describe ∏ S_{n}.
(Recall that 4={0,1,2,3}.)
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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=I_{B}.(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.
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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). | |