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 w
Solutions Homework 9
Math 300, Spring 2016
1. Let ` be a line in R2 . Prove that R2 cannot be a subset of
Due Tuesday, May 31
at the start of class
[
`i , a union of a countable
iN
collection of lines
Solutions Homework 8
Math 300, Spring 2016
Due Wednesday, May 25
at the start of class
1. Prove that cfw_0, 1 N and N have equal cardinality by giving a bijection from one to
the other.
Proof. To prov
Solutions Homework 7
Math 300, Spring 2015
Due Wednesday, May 18
at the start of class
1. Explicity construct a bijection between the two intervals [1, 1] and [2, 5].
Typically, problems like these in
Solutions Homework 2
Math 300, Spring 2016
Due Wednesday April 13
at the start of class
1. Draw Venn Diagrams for:
(a) (A B) C
(b) (A B) C
(c) (A B C) (A B C)
(d) A B C
2. Prove that the implication P
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 t
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 p
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-Bernst
Math 300 Foundations of Higher Mathematics
Syllabus Spring 2014
Instructor: Ryan Broderick, Locy 203, [email protected]
Oce Hours: M 3-5, W 2-5, F 2-3, and by appointment
TAs Oce Hours: T 10-
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
Solutions Homework 4
Math 300, Spring 2016
Due Wednesday April, 27
at the start of class
1. If a b(mod n), then gcd(a, n) = gcd(b, n).
Proof. Let d1 = gcd(a, n) and d2 = gcd(b, n). We will show that d
Solutions Homework 5
Math 300, Spring 2016
Due Wednesday May 4
at the start of class
1. Prove that for each natural number n,
3
3
3
1 + 2 + + n =
n(n + 1) 2
2
.
Proof. We will prove this statement by
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 a
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
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,
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
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 gue
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
Homework 3 Solutions
Math 300, Spring 2016
Due Wednesday April, 20
at the start of class
1. Is there any difference between the following statements? Explain!
(a) (x Z)(y Z)(3x = y).
(b) (y Z)(x Z)(3x
Solutions Homework 6
Math 300, Spring 2016
Due Wednesday, May 11
at the start of class
1. Find the number of ways to tile a 2 n grid for n 1 with a domino and prove that
your formula holds for all n 1
Homework 1 Solutions
Due Wednesday, April 6
Math 300, Spring 2016
at the start of class
1. Are the following statements true or false? Explain.
(a) R3 R3 True. Any set is a subset of itself.
(b) R2 R3
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 i
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
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 Poss
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2. Dene a relation R on P(Z+) by ARB if and only if A and B have the same smallest
K element.
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