Assignment
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Assigned
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Due
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Problems
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| HW1 |
8/28/13
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9/11/13
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Read
pages 1-15.
1. Do Exercise 1.3.1.
2. Do Exercise 1.3.3.
3. Do Exercise 1.3.6.
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| HW2 |
9/13/13
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9/20/13
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Read
pages 17-37.
1. Do Exercise 2.2.8(a) and first claim of 2.2.8(b).
2. Do Exercise 2.3.6(a).
3. Do Exercise 2.4.1.
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| HW3 |
9/20/13
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9/25/13
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Read
pages 39-45.
1. How many partial orderings on {0,1,2} are there?
(You may assume anything about counting that you learned in grade school,
even if haven't proved it yet within ZFC.)
2. Recall that a partial ordering of a set is a binary relation.
What are the least and largest number of pairs
that can occur in a partial 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?)
3. Do Exercise 3.2.1.
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| HW4 |
9/27/13
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10/4/13
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Read
pages 46-54.
1. Do Exercise 3.2.6.
2. Do Exercise 3.2.8.
3. Prove that
mn+k = mn · mk.
(You may need to prove some lemmas first.)
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| No HW |
10/2/13
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10/9/13
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Read
pages 65-68.
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| No HW |
10/9/13
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10/16/13
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No hw.
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| HW 5 |
10/16/13
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10/23/13
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Read
pages 69-73.
1. (Reworded.) Recall that we defined
tr.cl.(R) = ∪ ran(f)
where f is defined recursively by f(0) = R and
f(n+1) = R ° f(n). We showed that any transitive
binary relation containing R also contains
tr.cl.(R). Show here that
tr.cl.(R) is transitive.
2. Show that the function
F:ω×ω→ω
defined by
F(m,n) = 2m*(2n+1) - 1 is a bijection.
Conclude that |ω×ω| = |ω|.
(You may use any true fact about the arithmetic of the natural numbers
that you learned in grade school.)
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.)
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| HW 6 |
10/25/13
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10/30/13
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Read
pages 74-78, 90-97.
1.
Show that |2ω|=
|ωω|=|2ω×ω|.
2.
Show that the set of all equivalence relations
on ω has the same cardinality as
2ω.
3. Show that the set of all linear orders
on ω has the same cardinality as
2ω.
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| HW 7 |
11/1/13
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11/6/13
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Read
pages 103-122.
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.
Find all transitive sets of size 4.
3. Do
Exercise 6.2.8.
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| HW 8 |
11/8/13
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11/13/13
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Read exercises 3.3-3.5, which describe the recursive definition
of the hereditarily finite sets and the properties satisfied by
this set. That is, Vω satisfies most of the axioms
of ZFC.
1.
Do Exercise 6.5.3.
2.
Do Exercise 6.5.9.
3. Do
Exercise 6.5.11(a)(b)(d).
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| HW 9 |
11/15/13
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11/20/13
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Read 124-136.
1.
Do the part of Exercise 7.1.2 that concerns products of ordinals.
2.
Do Exercise 7.1.4.
3. Do
Exercise 7.1.5.
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| HW 10 |
11/22/13
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12/6/13
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Read 137-146.
1.
Exercise 7.2.6.
1.
Exercise 8.1.10.
1.
Exercise 8.1.16.
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