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MATH 4730/5730: Set Theory,
Fall 2023
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Homework
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Latex guide
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Assignment
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Assigned
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Due
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Problems
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| HW1 |
9/6/23
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9/13/23
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Read
Sections 1.1-2.2.
1. Do Exercise 1.3.1. (Exercise A.B.C means: Chapter A, Section B,
Exercise B.C.)
2. Do Exercise 1.3.6.
3. Do Exercise 2.2.8(a) and the first part of 2.2.8(b).
Solution sketches (setshw1sol.tex),
Solution sketches (setshw1sol.pdf).
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| HW2 |
9/14/23
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New Due Date!
9/22/23
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Read
pages 29-42. (That is, up to and including Section 3.1.)
Think about the following, but do not turn them in:
Exercises 2.3.6(a), 2.3.8, 2.4.1.
1. How many equivalence relations
on the set $\{0,1,2\}$ are there?
How many partial orderings on $\{0,1,2\}$ are there?
(To answer this, just write them down or draw
the appropriate picture. You don't have to prove
that your lists are complete, but to get full credit
your lists must be complete.
You may assume anything about counting that you
learned in grade school, even if we haven't proved it yet.)
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.
Solution sketches (setshw2sol.tex),
Solution sketches (setshw2sol.pdf)
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| HW3 |
9/22/23
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New Due Date!
9/29/23
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Read
pages 42-64.
Think about the following, but do not turn them in:
Exercises 3.1.1, 3.2.2, 3.2.7.
1. Do Exercise 2.3.9 (a). (If you use the hint,
show that it is true. Definition 2.3.13 is relevant to this problem.)
2. Do Exercise 3.2.6.
3. Do Exercise 3.2.8.
Solution sketches (setshw3sol.tex),
Solution sketches (setshw3sol.pdf)
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| HW4 |
9/29/23
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10/4/23
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1. Prove that $(m+n)\cdot k = (m\cdot k)+(n\cdot k)$
for all $m,n,k\in \mathbb N$.
2. Prove that $m\cdot n = n\cdot m$
for all $m,n\in \mathbb N$.
3. Prove that $m^{n+k}=m^n\cdot m^k$.
(You might need to prove some lemmas first.)
Solution sketches (setshw4sol.tex),
Solution sketches (setshw4sol.pdf)
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| HW5 |
10/11/23
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10/18/23
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Read
pages 65-79.
1. Define a binary operation $\circ$ on $\omega$ as follows.
Given $m, n\in\omega$, choose sets $A, B$ with $|A|=m$, $|B|=n$
and define $m\circ n = |A\times B|$.
(a) Show that $\circ $ is well defined.
(b) Show that $m\circ 0=0$ and $m\circ S(n)=(m\circ n)+m$.
(That is, $\circ$ satisfies the recursion that defines multiplication.)
(c) Conclude that $m\circ n = mn$. (This shows that
$|A\times B|=|A|\cdot |B|$ for finite sets.)
2. Show that the real line has the same cardinality as the real plane.
(Hint: By the CBS 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.).
3. Show that $|(A^B)^C|=|A^{B\times C}|$.
Solution sketches (setshw5sol.tex),
Solution sketches (setshw5sol.pdf)
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| HW6 |
10/20/23
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New Due Date!
10/27/23
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Read pages 93-102.
1. Let $A$ be an amorphous set.
Show that $A\times A$ is not amorphous.
Conclude that $A$ is an infinite set
such that $|A\times A|\neq |A|$.
2. Do Exercise 5.1.7.
3.
(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.
Solution sketches (setshw6sol.tex),
Solution sketches (setshw6sol.pdf)
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| HW7 |
10/27/23
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New Due Date!
11/3/23
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Read pages 103-123.
Read Exercises 6.1.1, 6.1.4, 6.2.7, 6.3.5
1. Working in ZF, show that an infinite well-orderable set
is Dedekind infinite. Conclude that an amorphous set
is not well-orderable. (Interpret ``well-orderable'' to
mean ``equipotent with an ordinal''.)
2. Do Exercise 6.1.3.
3. Do Exercise 6.2.8.
Solution sketches (setshw7sol.tex),
Solution sketches (setshw7sol.pdf)
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| No HW! |
11/1/23
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Nothing due on 11/8/23!
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Read Exercises 6.2.8, 6.5.1, 6.5.3, 6.5.9, 6.5.11, 6.5.12, 6.5.16.
Think about this problem: Is it possible to totally order
an amorphous set? (No need to turn in the answer.)
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| HW8 |
11/9/23
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New Due Date!
11/17/23
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Read Section 7.1.
1. Do Exercise 7.1.3.
2. Do Exercise 7.1.4.
3. Do Exercise 7.1.5.
Solution sketches (setshw8sol.tex),
Solution sketches (setshw8sol.pdf)
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| HW9 |
11/18/23
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New Due Date!
12/1/23
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1. Do Exercise 8.1.9.
2. Do Exercise 8.1.16.
3. Suppose you are working in a universe of sets satisfying ZF
in which all proper classes have the same size in the following
sense: whenever $\mathcal C$
and $\mathcal D$ are proper classes, then there is a class
bijection $F:{\mathcal C}\to {\mathcal D}$.
Show that the Axiom of Choice holds in your universe.
Solution sketches (setshw9sol.tex),
Solution sketches (setshw9sol.pdf)
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| HW10 |
12/3/23
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Unusual Due Date!
12/10/23
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Read Theorems 8.1.7, 8.1.10 and Exercise 8.1.8.
1. Using Zorn's Lemma, show that every
connected graph has a spanning tree.
2. Do Exercise 9.1.10.
3. Show that if $\alpha$ is any ordinal, then there
is an ordinal $\beta$ of countable cofinality
satisfying $\beta>\alpha$.
Solution sketches (setshw10sol.tex)
Solution sketches (setshw10sol.pdf)
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