Math 23b Homework 8 Solutions
1
1. (a) Let an = . Given > 0, we have by the Archimedean Property that there is
n2
1
1
some n >
so that n2 > and | n2 0| = n2 < . Hence, limn an = 0.
Let bn = n. If bn L then there exists N such that for all n N , |n L| =
|b
MATH 23b Homework 8
Sections 6.1-6.2
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must be written
MATH 23b Final Practice Exams
This exam will be closed-book, i.e., no books, notes, calculators or internet-connected devices
are allowed on this exam.
All your answers must be written in complete sentences.
The final exam is cumulative.
OVER for Practice
Name:
Final Exam
Math 23b, Fall 2015
I. True-False (2 points each) Answer these questions on this sheet.
Circle all true statements. No explanations are necessary.
Let S = cfw_n Z | 2n + 1 is divisible by 3 and T = cfw_n Z | gcd(n, 6) = 1.
(1) 10 (S T )c
MATH 23b Homework 5
Sections 2.2 & 3.1
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must be writt
MATH 23b Quizzam 1 Practice Test #3
You should give yourself 50 minutes to do this practice test (no books, notes, etc.).
1. For a nonzero integer n, define the set Mn = cfw_kn | k Z. Decide whether the
following statements are true for any nonzero intege
MATH 23b Homework 7
Chapter 5
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must be written in com
MATH 23b Quizzam 1 Practice Test #2
This practice test also has more questions than the actual quizzam will have, but I want
to give you extra practice. So you should also give yourself 80 minutes to do this one too.
Again, I recommend that you take this
MATH 23b Quizzam 1 Practice Test #3
SOLUTIONS
1. For a nonzero integer 72, dene the set Mn 2 cfw_kn | k E Z. Decide whether the
following statements are true for any nonzero integers a and b. If so, briey explain
Why. If not, give a counterexample.
(a) If
MATH 23b Quizzam 1 Practice Test #1
SOLUTIONS
1. Find the negation of the following statements. Which is true, the original statement
or its negation? (N o explanation necessary.)
(a) Suppose a, b and c are integers. If a divides c and b divides c, then a
An Inquiry-Based
I NTRODUCTION TO P ROOFS
A NSWERS TO E XERCISES
Jim Hefferon
version 1.0
page ii
Answers: Introduction to Proofs
N OTATION
N
Z, Z+
R
Q
a |b
a mod b
a c (mod b)
gcd(a, b), lcm(a, b)
aA
AB
A
Ac
A B, A B
A B, A B
|A|
P (A)
x 0 , x 1 , . . .,
An Inquiry-Based
I NTRODUCTION TO P ROOFS
Jim Hefferon
version 1.0
N OTATION
N
Z, Z+
R
Q
a |b
a mod b
a c (mod b)
gcd(a, b), lcm(a, b)
aA
AB
A
Ac
A B, A B
A B, A B
|A|
P (A)
x 0 , x 1 , . . ., (x 0 , x 1 )
lh(x 0 , x 1 , . . .)
A 0 A 1 A n1 , A n
f : D C
Practice Exam # 1
1. We dene a relation ~ on N by m N n if and only if 3 divides m + n.
(a) Is this relation an equivalence relation on N? For each condition, you must either
prove it holds or give a counterexample.
(b) Is this relation a partial order on
MATH 23b Quizzam 1 Practice Test #2
This practice test also has more questions than the actual quizzam will have, but I want
to give you extra practice. So you should also give yourself 80 minutes to do this one too.
Again, I recommend that you take this
Practice Exam # 3
9. Consider the relation ~ on N dened by m N n if and only if n = 2km for some integer
k 2 0.
(a) Show that this is a partial order on N. (You must prove each condition.)
(b) Is this relation a total order? If so, prove it. If not, give
gem/Ho (is
Name:
Final Exam
Math 23b, Fall 2015
I. True-False (2 points each) Answer these questions on this sheet.
Circle all true statements. No explanations are necessary.
0 Let S = cfw_n E Z I 2n +1 is divisible by 3 and T = cfw_n E Z | gccl(n, 6) = 1
MATH 23b Homework 1
Sections 1.1, 1.2
Except for problems 1a and 3, all your answers must be written in complete sentences.
1. Recall that an integer p > 1 is prime if it is only divisible by itself and 1. For a natural
number n N, define the set
Dn = cfw
MATH 23b Quizzam 2 Practice Exams
This quizzam is closed-book, i.e., no books, notes, calculators or internet-connected devices
are allowed on this quizzam.
All your answers must be written in complete sentences.
Theorems:
You may use the following theore
Math 23b
Homework 6
Spring, 2009
Due Monday, March 30.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem does not
Anna Medvedovsky
medved@brandeis.edu
Math 23b / Spring 2009
HW #3 solutions
1. For polynomial functions f (x) = x 1 and g (x) = x2 1 nd f g and g f .
Computing,
f g (x) = x2 2
and
g f (x) = x2 2x.
2. Explain why multiplication by 2 denes a bijection from
Math 23b
Homework 5
Spring, 2009
Due Wednesday, March 18.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem does
Math 23b
Homework 4
Spring, 2009
Due Wednesday, March 4.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem does n
Anna Medvedovsky
medved@brandeis.edu
Math 23b / Spring 2009
HW #3 solutions
1. Let A, B R, P = R>0 , f : R R. Write sentences negating the statements below.
a) Negation: There exists an x A such that for all b B , we have b x.
b) Negation: For all x A the
Math 23b
Homework 3
Spring, 2007
Due Thursday, February 12.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem doe
Math 23b
Homework 2
Spring, 2009
Due Wednesday, February 4.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem doe
Anna Medvedovsky
medved@brandeis.edu
Math 23b / Spring 2009
HW #1 solutions
1. A = cfw_j 2 j : j Z and B = cfw_k 2 + k : k Z, k 0. Prove that A = B .
First, I claim that A is a subset of B : If a A, then there exists j Z with j 2 j = a.
If j 1, then j 1 Z
Math 23b
Homework 7
Spring, 2009
Due Wednesday, April 8.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem does n
MATH 23b Quizzam 1 Practice Test #1
This is the midterm exam from last semester, for which students had 80 minutes. I recommend that you take this like an actual test (no books, notes, etc., and give yourself 80
minutes). Only check the solutions after yo
MATH 23b Homework 9
Sections 6.3-6.4
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must be written
MATH 23b Homework 4
Sections 2.1 & 2.2
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must be writt
MATH 23b Homework 10
Sections 8.1, 8.3, & 8.4
Instructions:
You may cite results proven in our textbook but not from other sources.
If you do use a result from the book, you must cite it by number, e.g., By Theorem
2.1.17, we know.
All your work must b