Set1
DUE DATE: Sep 12
1.1: In class, we considered the following "Railroad Problem": [A one-mile long railroad is
firmly fixed at both ends. During the night, a prankster cuts the rail and welds and additional
foot, causing the rail to bow up in the arc
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
December 16, 2010
1
HOMEWORK 13
Exercise 1.1. Establish the proper divergence of the following sequences.
Proof.
(a) Let an := n for all n N. Fix M > 0. By the Archimedian Property, there exists N N s
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
December 9, 2010
1
HOMEWORK 12
Exercise 1.1. Let cfw_fn be the Fibonaccia sequence of Example 3.1.2(d) (Bartle and Sherbert)
n=1
and let xn := fn+1 /fn . Given that limn xn = L exists, determine the
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
December 8, 2010
1
HOMEWORK 11
Exercise 1.1. Let x1 := 8, xn+1 = 1 xn + 2 for any n N. Show that cfw_xn is bounded and
n=1
2
monotone. Find the limit.
Proof.
First, suppose that cfw_xn is in fact bo
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
December 2, 2010
1
HOMEWORK 10
Exercise 1.1. Let yn :=
Find their limits.
n+1
n for n N. Show that cfw_yn and cfw_ nyn n=1 converge.
n=1
Proof.
For n N,
yn =
n+1
n=
n+1
n
n+1+ n
n+1+ n
(n + 1) n
1
=
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
November 18, 2010
1
HOMEWORK 9
Exercise 1.1. Let xn = 1/ ln(n + 1) for n N
(a) Use denition of limit to show that limn xn = 0
(b) Find a specic value of N as required in the denition of the limit of a
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
October 11, 2010
1
HOMEWORK 8
Exercise 1.1. Let X = Y = (0, 1). Dene h : X Y R by h(x, y ) = 2x + y .
(a) For each x X , nd f (x) := sup cfw_h(x, y ) : y Y and then inf cfw_f (x) : x X .
(a) For each
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
October 28, 2010
1
HOMEWORK 7
Exercise 1.1. Let S := cfw_1/n 1/m : n, m N. Find inf S and sup S .
Proof.
First note that for any m N that n > 0 implies that 1/n > 0. Together with n 1, we have that
1
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
October 21, 2010
1
HOMEWORK 6
Exercise 1.1. Let , > 0 and a R. Show that V (a) V (a) and V V (a) are -neighborhoods
of a for appropriate values of .
Proof.
Fix , > 0 and a R. Recall V (a) = cfw_x R :
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
October 14, 2010
1
HOMEWORK 5
1
Exercise 1.1. Prove that ( 2 (a + b)2 1 (a2 + b2 ) for all a, b R with equality i a = b.
2
Proof.
Fix a, b R. Then
12
1
(a + b)2 =
a + b2 + 2ab
4
4
12
a + b2 + 2|a| |b|
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331Midterm Exam
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-225/
01 November 2010
CMSC 331 Midterm Exam, Fall 2010 b
Name: _
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You will have seventy-five (75) minutes to complete this closed book/notes exam. Use the backs of
pages if you need
1 40/
2 30/
331Midterm Exam
3 45/
4 30/
5 30/
6 30/
7 20/
-225/
01 November 2010
CMSC 331 Midterm Exam, Fall 2010 a
Name: _
UMBC username:_
You will have seventy-five (75) minutes to complete this closed book/notes exam. Use the backs of
these pages if yo
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2 15/
331Midterm Exam
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4 35/
5 30/
6 20/
7 15/
-200/
05 April 2010
CMSC 331 Midterm Exam Fall 2010
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You will have seventy-five (75) minutes to complete this closed book/notes exam. Use the backs of
these pages if you need
Numerical Solution to Ordinary Differential Equations
By Gilberto E. Urroz, September 2004 In this document I present some notes related to finite difference approximations and the numerical solution of single and systems of ordinary differential equation
Examples of Initial-Value ODE Problems
By Gilberto E. Urroz, September 2004 The following are examples of initial-value ODE problems (IVP) solved using Matlab predefined functions ode23 and ode45. Example 1 Free-falling sphere in a fluid Script: Falling_S
Solution of non-linear equations
By Gilberto E. Urroz, September 2004
In this document I present methods for the solution of single non-linear equations as well
as for systems of such equations.
Solution of a single non-linear equation
Equations that can
Study Guide - Group Theory
MT 423A-01 - Abstract Algebra
Fall 2009
1. Know how to compute in permutation groups.
(a) What is the order of.
i. (123)(4567)
ii. (124)(3456)
iii. (13)(123)(234)(1435)
(b) Are the following even or odd?
i. (1324569)
ii. (123)(3
Study Guide - Intro to Groups
MT 423A - Abstract Algebra
Fall 2009
Chapter 0:
Know the Division Algorithm, the denition of gcd(m,n), and the gcd is a
linear combination theorem, and how to use them.
Be familiar and comfortable with the proof
a mod n = b
MATH 411
HOMEWORK 3
SOLUTIONS
7.4
25. Let G be a group and let Aut G be the group of automorphisms of G. Let
Inn G be the set of all inner automorphisms of G. Prove that Inn G is a
subgroup of Aut G.
Solution. Let f, g Inn G. So f (x) = a1 xa and g (x) =
Math 4124
Wednesday, January 26
First Homework Solutions
1. 1.1.9(b) on page 22 Let G = cfw_a + b 2 R | a, b Q. Prove that the nonzero
elements of G form a group under multiplication.
If x, y G then we may write x = + b 2 and y = c + d 2 where a, b, c, d
1 2 3 4 5 6 7 8 9
25/ 20/ 30/ 40/ 15/ 15/ 15/ 15/ 15/
331Midterm Exam
27 October 2008
CMSC 331 Midterm Exam Fall 2008
Name: _ UMBC username:_
200/
You will have seventy-five (75) minutes to complete this closed book/notes exam. Use the backs of these page
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
October 5, 2010
1
HOMEWORK 4
Exercise 1.1. Prove that the collection of all nite subsets of N is countable.
Proof.
Dene
F (N) = cfw_A N : A is nite .
Recall Nm = cfw_1, 2, . . . , m for any m N.
CLAIM
M361K: HOMEWORK 1 SOLUTIONS
LOUIZA FOULI
1. S ECTION 1.1 Problem 2: Prove the second De Morgan Law [Theorem 1.1.4(b)] If A, B and C are sets, then (b) A \ (B C) = (A \ B) (A \C) Proof. We have to show that A \ (B C) (A \ B) (A \C) and that (A \ B) (A \C)
1
Section 3.3 - 1, 2, 3, 4
1. 1. Let x1 := 8 and xn+1 := 1 xn + 2 for n N . Show that (xn ) is 2 bounded and monotone. Find the limit. Proof. First, lets show that it is monotone (decreasing). Note that x1 = 8 > x2 = 6. For induction, assume that xk > xk+
1
Section 3.1 - 1d, 2b, 3b, 4, 5a, 5c, 8, 11
1. 1d. Write the first five terms of x :=
1 Proof. x1 = 2 , x2 = 1 , x3 = 5 1 10 1 n2 +2 . 1 17 , x5 1 26
x4 =
=
2. 2b. The first few terms of a sequence (xn ) are given below. Give a formula for the nth term x
1
Section 2.5 - 2, 7, 8, 9, 17
1. 2. If S is nonempty, show that S is bounded if and only if there exists a closed bounded interval I such that S I. Proof. () If S is bounded, then there exists u = sup(S) and v = inf(S). Define the interval I := [v, u] an
1
Section 2.4 - sup portion of 4(a,b), and 6, 13, 18
1. 4. (a) Let a > 0 and let aS := cfw_as : s S. Prove sup(aS) = a sup S. Proof. Let u = sup(S). This means that u is an upper bound of S. Hence, s u for all s. Since a > 0, as au for all s. This means a
1
Section 2.2 - 1(b), 4, 6(a), 15, 16(a)
1. If a, b Proof. |a/b| = |a 1/b| = |a| |1/b| defn. of division 2.2.2(a) and b = 0. Show |a/b| = |a| / |b|
Now the question we must examine is: Does |1/b| = 1/ |b|? The answer is yes and is justied by using the den
1
Section 2.1 - 10, 13, 14, 20, 22
1. (a) If a < b and c < d, prove that a + c < b + d. Proof. We will show two cases: i. c < d To show this, if a < b, then by 2.1.6(a), b - a P . Likewise, if c < d, then d - c P . By 2.1.5(i), (b - a)+)d - c) P . This im