How to Construct a Pair Presentation
Math 300: Foundations of Higher Mathematics
Below is a list of things that you and your partner should do before meeting with me to go
over your presentation. In the pair presentation one partner will be speaking while
Math 300: Worked Examples
Northwestern, Spring 2013
This is a collection of examples and proofs worked out in detail. I also try to give a sense of the
thought process you might go through in trying to come up with some of these proofs. Remember
that a la
1. State Cantors Theorem, the Continuum Hypothesis and the Schroeder-Bernstein Theorem.
2. Provide the following denitions:
(a) A function f : A B is surjective if .
(b) A relation R on a set A is a partial ordering if.
(c) Two sets A and B are numericall
1. State Cantors Theorem, the Continuum Hypothesis and the Schroeder-Bernstein Theorem.
Cantors Theorem. For any set A, A
P (A).
Continuum Hypothesis. There is no set A satisfying Z
A
Schroeder-Bernstein Theorem. If A and B are sets, A
A B.
R.
B and B
A,
Math 300 Foundations of Higher Mathematics
Syllabus Spring 2014
Instructor: Ryan Broderick, Locy 203, ryan@math.northwestern.edu
Oce Hours: M 3-5, W 2-5, F 2-3, and by appointment
TAs Oce Hours: T 10-12, W 12-1, and by appointment
Textbook: Bond & Keane,
Name:
Northwestern UniversityStudent ID:
Math 300 Final Exam
Spring Quarter 2013
Tuesday, June 11, 2013
Put a check mark next to your section:
Tsygan
Khorami
Question Possible Score
points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
TOTAL
100
Canez
Math 300: Final Exam Practice
Here are some problems to think about when preparing for the nal exam. These together with
homework problems, both assigned and optional, and quizzes should give a good idea of what to
expect. In the solutions I will give an
Math 300: Final Exam Practice Solutions
1. Let A be the set of all real numbers which are zeros of polynomials with integer coecients:
A := cfw_ R | there exists p(x) = an xn + + a1 x + a0 with all ai Z such that p() = 0.
Show that A is countable. (Elemen
Name:
Northwestern UniversityStudent ID:
Math 300 Midterm 1
Spring Quarter 2013
Wednesday, May 1, 2013
Put a check mark next to your section:
Tsygan
Khorami
Canez
Broderick
Instructions:
Question Possible Score
points
1
10
2
10
3
15
4
20
5
10
6
10
7
10
8
Math 300: Quiz 1 Solution
Northwestern, Spring 2013
The following is the precise denition of what it means for a function f (x) to be continuous (all
numbers involved are real):
For all a and all
> 0 there exists > 0 such that |f (x) f (a)| <
if |x a| < .
Math 300: Quiz 3 Solution
Northwestern, Spring 2013
For any a R let La denote the line y = ax in R2 . Compute aR La and aR La , and prove
that your answers are correct.
First let us come up with a guess as to what this intersection and union actually are.
Math 300: Quiz 6
Northwestern, Spring 2013
May 28, 2013
Name:
Consider the set of all polynomials of the form p(x) = ax2 + bx + c with a, b, c Z. Take it as
a given that this set is countable. Now consider the set A of all real numbers which are zeros of
Math 300: Quiz 4
Northwestern, Spring 2013
May 7, 2013
Name:
Let R denote the Cartesian product of innitely many copies of R:
R = R R R = cfw_(x1 , x2 , x3 , . . .) | each xi is in R
innitely many times
To be clear, this is analogous to Rn only that each
Math 300: Quiz 3
Northwestern, Spring 2013
April 23, 2013
Name:
For any a R let La denote the line y = ax in R2 . Compute
that your answers are correct.
aR La
and
aR La ,
and prove
Math 300: Quiz 2
Northwestern, Spring 2013
April 16, 2013
Name:
Recall that Q denotes the set of rational numbers. Prove that r Q | r 5 is rational = cfw_0.
You may use the fact that 5 is irrational without proof.
Math 300: Quiz 1
Northwestern, Spring 2013
April 9, 2013
Name:
The following is the precise denition of what it means for a function f (x) to be continuous (all
numbers involved are real):
For all a and all
> 0 there exists > 0 such that |f (x) f (a)| <
i
Math 300: Quiz 2 Solution
Northwestern, Spring 2013
Recall that Q denotes the set of rational numbers. Prove that r Q | r 5 is rational = cfw_0.
You may use the fact that 5 is irrational without proof.
In plain english, all this claim says is that 0 is th
Math 300: Quiz 5 Solution
Northwestern, Spring 2013
Recall that Q denotes the set of nonzero rational numbers. Dene the binary operation on
by a b = a . Show that is not associative, not commutative, and does not have an identity
b
element. Also, give an
Math 300: Quiz 4 Solution
Northwestern, Spring 2013
Let R denote the Cartesian product of innitely many copies of R:
R = R R R = cfw_(x1 , x2 , x3 , . . .) | each xi is in R
innitely many times
To be clear, this is analogous to Rn only that each element h
Math 300: Final Exam Solutions
Northwestern, Spring 2013
1
1
1. (a) For all N Z+ there exists > 0 such that for all n, m N , n m or n m.
(b) There exists > 0 such that for all > 0 there exists x, y such that |x y| < and
|x5 y 5 | . (This statement is the
Math 300: Midterm 1 Practice
Here are some problems to think about when preparing for the midterm. These in addition to
homework problems should give an idea of what to expect. The problem from the second quiz is
good too, but the problems from the rst an
Page 3 of 7
2. Dene a relation R on P(Z+) by ARB if and only if A and B have the same smallest
K element.
/
(a) Prove that R is an equivalence relation.
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Theory of Property Rights paper
Health Care and Price Control
In a perfect market, 6 assumptions that are made:
Perfect competition
Perfect information
Free entry and exit
Everyone acts in rational self interest
Homogenous products sold
No transaction cos