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
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
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 sen
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 divisib
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., B
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 t
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
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
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, b
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
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
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 ,
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
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
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
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
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 a
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 complet
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 th
Anna Medvedovsky
[email protected]
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.
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
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
Anna Medvedovsky
[email protected]
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
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 pro
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 pro
Anna Medvedovsky
[email protected]
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, t
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