183
Chapter 13:The Real Numbers
SOLUTIONS FOR PART IV
184
and S contains sup(S) and
13.8. If S is a bounded set of real numbers,
inf (S), thenS is a closed interval FALSE. Counterexamples include the
nite setS = cfw_
0, 1and the uncountable set
S = [0, 1]
MATH 348 Section C1, Writing Assignment 4
*Xinhe Geng 678388888*
In this assignment, we solve the following problem:
Problem 4.20
Let f : R2 R2 be defined by f(x,y) = (ax - by, bx + ay), where a,b are
numbers with a2 + b2 6= 0.
a) Prove that f is a biject
MATH 348 Section C1
Homework 2
Due Wednesday, September 9
Homework assignment
Problems: 2.4 (a,c,e), 2.10 (a,d,h), 2.21, 2.24, 2.28, 2.34, 2.44, 2.53
from Chapter 2 in the textbook.
Writing assignment
Use the method of proof by contradiction to prove th
Solution
October 26, 2015
Midterm 2
MATH 348 Section C1
Instructions: There are 5 problems, worth 10 points each. Read all questions carefully. If
a proof is required, you must write your proof in a clear and rigorous manner. The proof
should be brief, y
MATH 348 Section C1
Homework# 3 Solution
Homework assignment
3.1 P (n) : n < 100.
3.6 False. Counterexample: P (n) : n > 1. (P (1) is not true, but P (n) is true for all n 2.)
P
3.17 Let P (n) denote ni=1 i(i + 1) = n(n+1)(n+2)
.
3
Basis step: Since 1 (1
MATH 348 Section C1
Homework# 10 Solution
Homework assignment
7.9 By the Fermats little theorem, 212 1 mod 13. So
2100 (212 )8 24 1 16 3
mod 13.
7.17 b) Since 5 1 mod 6, we have that
n3 + 5n n3 n n(n2 1) n(n 1)(n + 1)
mod 6.
For each n Z, cfw_n, n 1, n +
Sample Final Exam Solution
MATH 348 Section C1
1. Determine whether each of the following statements is TRUE or FALSE.
(a) Let f : A B and g : B C be functions. If g f is injective, then g is injective.
False.
(b) A countable union of countable sets is co
Solution
September 28, 2015
Midterm 1
MATH 348 Section C1
Instructions: There are 5 problems, worth 10 points each. Read all questions carefully. If
a proof is required, you must write your proof in a clear and rigorous manner. The proof
should be brief,
MATH 348 Section C1
Homework 4
Due Wednesday, September 23
Homework assignment
Exercises: 4.9, 4.12, 4.23, 4.25(a,b,d), 4.26, 4.31, 4.34, 4.35 from Chapter 4 of the textbook.
Writing assignment
Exercise 4.20 from Chapter 4 of the textbook. Be sure to gi
MATH 348 Section C1
Homework 3
Due Wednesday, September 16
Homework assignment
Problems: 3.1, 3.6, 3.17, 3.28, 3.31, 3.41, 3.49 (a,c), 3.56 from Chapter
3 in the textbook.
Writing assignment
Use induction to solve the following problem. Be sure to give
Sample Final Exam
MATH 348 Section C1
1. Determine whether each of the following statements is TRUE or FALSE.
(a) Let f : A B and g : B C be functions. If g f is injective, then g is injective.
(b) A countable union of countable sets is countable.
(c) For
Sample Midterm 3
MATH 348
Instructions: Unless otherwise stated, you may use results learned in class to solve each
of these problems.
1. See textbook or lecture notes.
(a) Let R be an equivalence relation on a set S and x S. The equivalence class contai
MATH 348 Section C1
Homework# 7 Solution
Homework assignment
14.2 a) Let an = n.
Then an+1 an = n + 1 n =
1
n+1+ n
0.
b) Let an = n2 .
Then an+1 an = (n + 1)2 = n2 = 2n + 1, so lim(an+1 an ) does not exist.
c) Let an = nL, where L > 0.
Then an+1 an = (n
NAME: Solution
Sample Midterm 2
MATH 348
Instructions: Unless otherwise stated, you may use results learned in class to solve each
of these problems.
1. See lecture notes or the textbook for the solution to this problem.
(a) Give the precise definition o
MATH 348 Section C1
Homework 8
Due Wednesday, November 4
Homework assignment
Exercises: 7, 8, 13, 17, 24, 28 from Chapter 6.
Writing assignment
Write a complete solution to Exercise 6.41 (Polyas proof for infinitude
of primes).
You are required to type
MATH 348 Section C1
Homework 6
Due Wednesday, October 14
Homework assignment
Exercises: 26, 28, 29 from Chapter 13 and 1, 3, 8, 14 from Chapter 14.
Writing assignment
Write a complete solution to the following problem.
Prove, using the definition, that
MATH 348 Section C1, Writing Assignment 1
Xinhe Geng
678388888
In this assignment, we solve the following problem:
Prove that the Least Upper Bound Property (i.e., the Completeness Axiom) holds for R if and only if the Greatest Lower Bound Property
holds
MATH 348 Section C1
Homework# 1 Solution
Homework assignment
1.27 Let x be a real number. If x 6= 1, then |x + 1| > 0, so |x/(x + 1)| 1 is equivalent to
|x| |x + 1|.
(1)
Since |x| and |x + 1| are nonnegative, we have that the inequality (1) is equivalent
63
Part II Solutions
5. COMBINATORIAL REASONING
5.1. When rollingn dice, the probability
is/ 2
1 that the sum of the numbers
n
obtained is even.There are 6 equally likely outcomes;
we show that in
n1
half of them the sum is even.For each of the 6 ways to
NAME:
Sample Midterm 1
MATH 348
Instructions: The actual first midterm may not look like this, but it should at least give
you some idea of what you should prepare for the exam. For each problem, if a proof
is required, you must write your proof in a cle
Solution
November 30, 2015
Midterm 3
MATH 348 Section C1
Instructions: There are 5 problems, worth 10 points each. Read all questions carefully. If
a proof is required, you must write your proof in a clear and rigorous manner. The proof
should be brief,
1
Part I Solutions
SOLUTIONS FOR PART I
1. NUMBERS, SETS, AND FUNCTIONS
1.1. We have at least four times as many chairs as tables. The number of
chairs (c) is at least
( ) four times the number of tables
t). (Hencec 4t.
2
Chapter 1:Numbers, Sets, and Fun
MATH 348 Section C1
Homework# 4 Solution
Homework assignment
4.9 True. Consider 4 cases: (f is nondecreasing or nonincreasing) and (g is nondecreasing
or nonincreasing).
Assume that f, g are nondecreasing. Let x, y R with x > y. Since f is nondecreasing,
MATH 348 Section C1
Homework 11
Due Wednesday, December 9
Homework assignment
Exercise: 1.55 from Chapter 1 and Problems: 2, 3, 5, 10(b), 11(b), 12(b), 13(b) from
the sample final exam (the link can be found on Compass).
Writing assignment
There is no w
MATH 348 Section C1
Homework# 8 Solution
Homework assignment
6.7 Since (9, 15) = 3 and 3 does not divide 61, we cannot write 61 as an integer combination
of 9 and 15. On the other hand, (9, 16) = 1 and 1 divides 61, so 61 is an integer combination
of 9 an
MATH 348 Section C1
Homework 1
Due Wednesday, September 2
Homework assignment
Problems: 1.27, 1.29, 1.30, 1.34, 1.36, 1.41 (b,d,e) from Chapter 1 in
the textbook.
Writing assignment
Write a complete solution to the following problem. Be sure to give
a d
MATH 348 Section C1
Homework 10
Due Wednesday, November 18
Homework assignment
Exercises: 9, 17(b), 20, 24, 31, 34 from Chapter 7.
Writing assignment
Every book has a unique International Standard Book Number (ISBN). There are two
types of ISBNs, ISBN-1
MATH 348 FUNDAMENTAL MATHEMATICS
SECTION E1, SPRING 2014
RECURRENCE RELATIONS III
ASSIGNMENT 13
Student:
Date:
Problem: On the 1st of March, a bank loans a man $2500 at a xed rate of interest of
1.5% per month. The interest is added on the last day of eac