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MATH 4730/5730: Set Theory,
Fall 2019


Homework



Assignment

Assigned

Due

Problems

HW1 
9/4/19

9/11/19

Read
pages 128.
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).

HW2 
9/11/19

9/18/19

Read
pages 2942.
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 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.

HW3 
9/18/19

9/25/19

Read
pages 4264.
Think about the following, but do not turn them in:
Exercises 3.2.2, 3.2.7.
1. Do Exercise 3.2.6.
2. Do Exercise 3.2.8.
3. Prove that $m^{n+k}=m^n\cdot m^k$.
(You may need to prove some lemmas first.)

HW4 
9/25/19

10/2/19

Read
pages 6579.
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
11 functions in each direction. For a 11 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 $(A^B)^C=A^{B\times C}$.

HW5 
10/3/19

10/9/19

The exponentiation used in these problems is
defined so that $A^B=\{f: B\to A\;\;f\;\textrm{is a function}\}$.
1. Show that
$2^{\omega}=\omega^{\omega}=2^{\omega\times\omega}$.
2. Show that the set of all
equivalence relations on $\omega$ has the same cardinality as $2^{\omega}$.
3. Show that the set of all
linear orders on $\omega$ has the same cardinality as $2^{\omega}$.

HW6 
10/16/19

10/23/19

Read pages 93102.
1. Do Exercise 5.1.3.
2. Do Exercise 5.1.7.
3.
(a) Give an example of a set that is transitive, but not wellordered by epsilon.
(b) Give an example of a set that is wellordered by epsilon, but is not transitive.

HW7 
10/23/19

New Due Date!
11/1/19

Read pages 103123.
Read Exercises 6.2.7, 6.2.8, 6.3.5, 6.5.1, 6.5.12, 6.5.16.
1. Do Exercise 6.5.3.
2. Do Exercise 6.5.9.
3. Do Exercise 6.5.11(a)(b)(d).




Read pages 124134.

HW8 
11/7/19

11/13/19

1. Do Exercise 7.1.3.
2. Do Exercise 7.1.4.
3. Do Exercise 7.1.5.
Solutions.

HW9 
11/13/19

11/20/19

1. Do Exercise 8.1.9.
2. Do Exercise 8.1.16.
3. Suppose you are working in a universe of sets
that has the following property: 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.

HW10 
11/21/19

New Due Date! 12/6/19

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 $\beta>\alpha$.


