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MATH 4730/5730: Set Theory, Fall 2019


HW1 9/4/19
Read pages 1-28.

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
Read pages 29-42.

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
Read pages 42-64.

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
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 $|(A^B)^C|=|A^{B\times C}|$.
HW5 10/3/19
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 93-102.

1. Do Exercise 5.1.3.

2. Do Exercise 5.1.7.

(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.

HW7 10/23/19
New Due Date!
Read pages 103-123.

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 124-134.

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.


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!
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 $\beta>\alpha$.