MATH 3300
Midterm. February 12, 2014
Name:
Student Id:
There are 7 problems. Proofs are expected to be short; one short paragraph should be the average
length. You are not allowed to use notes, textbooks or electronic devices during the test. Please
write
MATH 3300. Problem set 2.
1. Prove that if f : A B and g : B C are surjective functions, then g f
is a surjective function.
2. Let f : A B and g : B A. Show that if g f is the identify function
on A, then f is injective and g is surjective.
3. A subset A
MATH 3300. Problem set 1.
1. Let A, B, C be nite sets. Use Venn-diagrams to justify the following
formula.
|A B C | = |A| + |B | + |C | |A B | |A C | |B C | + |A B C |
2. Let A, B and C be sets. Prove the following statement.
(A B ) C = (A C ) (B C )
3. L
MATH 3300. Problem set 1.
1. Let f : X Y be a function, A X and B Y .
(a) If f is injective prove that A = f 1 (f (A). Provide an example
where equality fails in the case that f is not injective.
Proof. First we prove that A f 1 (f (A). Suppose that x A.
MATH 3300. Problem set 3.
Recall that the statement A and B have the same cardinality is denoted by
A =c B in the lectures, by A B in the class notes, and by |A| = |B | in some
textbooks.
1. Let f : X Y be a function.
(a) Show that if f is injective, and
MATH 3300. Problem set 4.
Denition 1 (Countable Sets) A set A is countable if either A is nite or
there is a bijection N A. If a set is not countable, then it is called uncountable.
Theorem 2 Let A be a set. The following statements are equivalent.
A is
MATH 3300
Set Theory
Winter 2014
Slot
02
Section
001
Classroom
HH 3017
Instructor
Dr. Eduardo Mart
nez-Pedroza
email: [email protected]
Oce: HH-3006.
Oce hours: Mondays 10am-11am, and Wednesdays 11am-noon.
Textbook
Set Theory; Class notes. by Michael M.
MATH 3300. Problem set 6.
1. Let x, y, z be cardinal numbers.
(a) If x y prove that x + z y + z .
(b) Find an example showing that (a) does not hold if is replaced
by <.
(c) Prove that (a) is true if is replaced by =.
(d) Is the converse of (a) true? Just
MATH 3300. Problem set 5.
1. Use the fact that there are innitely many prime numbers to show that
for every n N there is an injective function Nn N.
2. For any three sets A, B, C , prove that C AB is equinumerous to (C B )A .
Suggestion: For any p : A B C
MATHEM ATICS 3300
SET THEORY
MATH 3300: Set Theory
This course has a somewhat more philosophical orientation then other mathematics offerings. The
following are some interesting examples of topics discussed.
1.
Consider the following problem:
In a certain