List of Homework Assignments

Assignments 1-6 are from Kaplansky's book, Set Theory and Metric Spaces. Assignments 7-? are based on excerpts from my book, Handbook of Analysis and its Foundations (HAF).

Asgt. no. Date due Description

7 Friday 6 Feb or Monday 9 Feb In HAF, do all the "verify" and "show" steps in the proof of Theorem 3.41. That is: (a) F(X₀) is a lower set in Y. (b,c) F gives an order isomorphism from X₀ onto F(X₀) -- that is, (b) one-to-one and (c) order-preserving. (d) In the last paragraph, show that the range of gof is a lower set of X.

6 Friday 30 Jan In the collection of all subsets of the integers (ordered by inclusion), show that the collection of all subsets that have 4 or fewer members is not a predecessor set or a principal lower set (though we did show in class that it is a lower set).

5 Friday 23 Jan Page 18 problem 15.

4 Wednesday 21 Jan Let L be the collection of all subsets of N that are finite or cofinite. Partially order L by "is a subset of". Show that L is a lattice, but not a complete lattice. (Hints given in class.)

3 Monday 19 Jan Show that the map taking any element to its principal lower set is an isomorphism from any poset onto a collection of subsets of a set.

2 Wednesday 14 Jan Page 18 problem 18.

1 Monday 12 Jan Page 3 problems 7 and 8, and pages 8-9 problems 5 and 8. Write proofs in complete sentences.

Asgt. no.	Date due	Description
7	Friday 6 Feb or Monday 9 Feb	In HAF, do all the "verify" and "show" steps in the proof of Theorem 3.41. That is: (a) F(X₀) is a lower set in Y. (b,c) F gives an order isomorphism from X₀ onto F(X₀) -- that is, (b) one-to-one and (c) order-preserving. (d) In the last paragraph, show that the range of gof is a lower set of X.
6	Friday 30 Jan	In the collection of all subsets of the integers (ordered by inclusion), show that the collection of all subsets that have 4 or fewer members is not a predecessor set or a principal lower set (though we did show in class that it is a lower set).
5	Friday 23 Jan	Page 18 problem 15.
4	Wednesday 21 Jan	Let L be the collection of all subsets of N that are finite or cofinite. Partially order L by "is a subset of". Show that L is a lattice, but not a complete lattice. (Hints given in class.)
3	Monday 19 Jan	Show that the map taking any element to its principal lower set is an isomorphism from any poset onto a collection of subsets of a set.
2	Wednesday 14 Jan	Page 18 problem 18.
1	Monday 12 Jan	Page 3 problems 7 and 8, and pages 8-9 problems 5 and 8. Write proofs in complete sentences.