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
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
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 := f
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
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.
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 v
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,
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 i
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 o
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. The
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
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. Exa
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.
Solut
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)
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.
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.
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 w
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 ni
MATH 0413 - Intro. Theoretical Mathematics
Ross Ingram
September 28, 2010
1
HOMEWORK 3
Exercise 1.1. Conjecture a formula for the sum 1 + 1/(1 3) + 1/(3 5) + . . . + 1/(2n 1)(2n + 1)
Prove conjecture
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
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 th
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 = su
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
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/
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,