Solutions Midterm Exam 2 Apr. 9, 2014
1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the
claim, derive a contradiction or give a counter-example).
(a) (10 points) If f : D R is continuous and D is closed,
Set theory, Spring 2015
Homework 1 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 1.1). If R is a relation, then we dene R1 =
cfw_(y, x) | x R y. Give an example where R is a function but R1 is not.
Build a rela
Set theory, Spring 2015
Homework 3 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 3.2). Let (A, A ) and (B, B ) be wellorderings.
Use Theorem 2.38 to prove that exactly one of the following conditions holds:
I.
Set theory, Spring 2015
Homework 4 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Note: In these solutions I am adopting the convention we used in class
that + = A () and = M (), where A and M are recursively
dened in the usual way. I
Set theory, Spring 2015
Homework 2 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 2.8). In general, dene
A
B = cfw_f | f is a function from A to B.
Consider a function of the form (a, b) S(a,b) with domain A B.
Set theory, Spring 2015
Homework 5 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 4.1). Let
1. Prove that
<
<
2=
n
n<
2.
2 is countable.
2. Let F = cfw_x n | n < | x 2. Prove that |F| = 20 .
3. Prove that there
Solutions Midterm Exam 1 Mar. 5, 2014
1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the
claim, derive a contradiction or give a counter-example).
(a) (10 points) If A B, and B is countable, then A is cou
Solutions Final Exam May. 14, 2014
1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the
claim, derive a contradiction or give a counter-example).
(a) (10 points) There exist open intervals In with In+1 In s
Final Exam Solutions
1. Let f (x) be a continuous function on [a,b] with f (a) < 0 < f (b).
a. (10 pts) Let x0 be such that f (x0 ) > 0. Show that there is a interval of the form
I = (x0 , x0 + ) such that f (x) f (x0 ) on I (a, b).
2
By the uniform conti
Set theory, Spring 2015
Homework 6 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1. Suppose that f : R R is any function. Show that there exist
two injective functions g : R R and h : R R such that f = g + h.
We x a bijection