Math 3000 Section 003 Intro to Abstract Math Midterm 2
Department of Mathematical and Statistical Sciences
University of Colorado Denver, Spring 2012
Solutions (April 20, 2012)
Problem
Minutes
Points
Score
1
5
9
2a
5
2b
10
5
2c
5
3a
5
3b
10
9
3c
4
4a
10
4

Problem 1. Let G be a group and let H, K be two subgroups of G such that H K
is also a subgroup. Prove that either H K or K H.
Solution: Suppose that neither H K nor K H. Then there is h H,
h 6 K and there is k K, k 6 H. Since H K is a subgroup, we have h

Math 417
Exam 1 (Solutions)
Prof. I.Kapovich February 27, 2009
Problem 1.[20 points]
For each of the following statements indicate whether it is true or false.
You DO NOT need to provide explanations for your answers in this problem.
(1) The set cfw_a : a

Mathematics 310
Robert Gross
Homework 3
Answers
1. Let
!
a b
2
2
a,
b
R,
a
+
b
=
6
0
.
G1 =
b a
Last week, we saw that G1 is an abelian group with group operation matrix
multiplication.
!
a b
Define a function : C G1 with the formula (a + ib) =
. Show

MATH 411/511, HOMEWORK #3
DUE TUESDAY, SEPT. 20 AT THE START OF CLASS.
1. Suppose a, b Z, and n Z+ . Prove that
a b (mod n)
if and only if a and b have the same remainder when divided by n.
2. Let c, n Z+ .
a. Show that if c and n are relatively prime, th

1
Solutions to assignment 3, due May 31
Problem 11.26 Use the Euclidean Algorithm to nd the GCD for each of the following
pairs of integers:
(a) 51 and 288
In this case, we write 288 = 5 51 + 33. Following through, we obtain
Solution:
51 = 1 33 + 18
33 =

Abstract Algebra I
Math 481, Fall 2014
Professor Ben Richert
Exam 1
Solutions
Problem 1 (10pts) Complete the following definitions:
(a4pts) A monoid consists of a . . .
Solution. nonempty set M and a binary operation : M M M with the following properties

MATH 411/511, HOMEWORK #1
DUE THURSDAY, SEPT. 8 AT THE START OF CLASS.
This assignment is based on material covered in class and Sections 1.2, 0.3, and 0.4 from
Nicholson. See also NTNotes.pdf.
1. For each of the pairs of integers, a and b, use the Euclid

Introduction to Numerical Analysis
Doron Levy
Department of Mathematics
and
Center for Scientific Computation and Mathematical Modeling (CSCAMM)
University of Maryland
June 14, 2012
D. Levy
CONTENTS
Contents
1 Introduction
1
2 Methods for Solving Nonlinea

The International Journal Of Engineering And Science (IJES)
|Volume|2 |Issue| 11|Pages| 05-12|2013|
ISSN(e): 2319 1813 ISSN(p): 2319 1805
On the Rate of Convergence of Newton-Raphson Method
Ranbir Soram1, Sudipta Roy2, Soram Rakesh Singh3, Memeta Khomdram

MA2501 Numerical
Methods
Spring 2015
Norwegian University of Science
and Technology
Department of Mathematics
Solutions to exercise set 7
1 Cf. Cheney and Kincaid, Exercise 4.1.9
Consider the data points
xi
f (xi )
0
1
1
9
2
23
4
93
6
259
a) Find the inte

Jim Lambers
MAT 460/560
Fall Semester 2009-10
Lecture 12 Notes
These notes correspond to Section 2.4 in the text.
Error Analysis for Iterative Methods
In general, nonlinear equations cannot be solved in a nite sequence of steps. As linear equations
can be

7/27/99
MATH 373 Section H6
Problem Set #2 Solutions
1. Solve the following via the quadratic formula: x2 + 106 x 106 . How will a computer
using single precision floating point arithmetic compute the solutions? How can we alter the
quadratic formula in t

NUMERICAL SOLUTIONS:
Solved Examples
By
Mahmoud SAYED AHMED
Ph.D. Candidate
Department of Civil Engineering, Ryerson University
Toronto, Ontario 2013
Table of Contents
Part I: Numerical Solution for Single Variable. 2
1.1.
Newton-Raphson Method . 2
1.2.
S

Numerical Analysis
Grinshpan
Linear Convergence
Let a positive sequence cfw_an converge to 0 and satisfy the condition
an+1
= C,
()
lim
n an
for some C 0.
Condition () implies that C 1. Indeed, since an 0, there exist infinitely many
indices n such that

MTH/CSC 421
Review 1
1. Let a = 0.1 and b = 1. Show that at least 27 steps of bisection method are needed to determine
the root of a function f (x), with an error of at most 12 108 .
Hint: Recall that error
ba
2n+1 .
2. For the bisection method, show that

Metric Spaces
Lecture Notes and Exercises, Fall 2016
M.van den Berg
School of Mathematics
University of Bristol
BS8 1TW Bristol, UK
mamvdb@bristol.ac.uk
1
Definition of a metric space.
Let X be a set, X 6= . Elements of X will be called points.
Definition

Section 5.2 Compound Interest
With annual simple interest, you earn interest each year on your original investment. With
annual compound interest, however, you earn interest both on your original investment and
on any previously earned interest.
EXAMPLE:

Rational Numbers
There are four standard arithmetic operations: addition, subtraction,
multiplication, and division.
Just as we took differences of natural numbers to represent integers, here the
essence of the process is to use ordered pairs representi

SOLUTIONS TO HOMEWORK # 2
3.5 (a) Prove |b| a a b a. Let us note that a 0, a 0.
We consider two cases.
(i) Let b 0. Then |b| = b and we have: |b| a b a a b a (it is
obvious that a b since b 0, a 0).
(ii) Let b < 0. Then |b| = b, and we have |b| a b a a b

Problem 1. Let f : A R R have the property that for every x A, there exists > 0 such that
f (t) > if t (x , x + ) A. If the set A is compact, prove there exists c > 0 such that f (x) > c for all
x A.
Proof:
For any x A, there exists x > 0 such that f (t)

Denker
Spring 2012
312 Analysis - Assignment 2
Solution
Problem 1:
1. For all non-negative x, there is some y real such that x = y 2 . True: take y :=
x.
2. There is some non-negative x for which
there is some y real such that x = y 2 . True,
take for exa

Denker
Spring 2010
312 Analysis - Assignment 4
Due Date: Monday, 02/15/2010 in class.
Solve as many problems as you can. Only one problem will be graded in detail. Full
credit for 3 solved problems. Any of the remaining problems may appear on a midterm
or

SOLUTIONS TO HOMEWORK # 3
4.6 (a) Let s S. Since inf S is a lower bound of S and sup S is an upper bound
of S, we have inf S s and s sup S. It remains to use the transitivity of the order.
4.6 (b) For each s S we have inf S s sup S; inf S = sup S implies

Denker
Spring 2010
312 Analysis - Assignment 5
Due Date: Monday, 02/22/2010 in class.
NOTE: All problems from assignment 15 may appear on the first exam on
Feb 26!
1
1
1
Problem 1: Let an = n+1
+ n+2
+ + 2n
. Show that the sequence (an )n1 is increasing
a

8
Personal Finance
Compound Interest
Annuities
Amortization and Sinking Funds
8.3
Simple Interest
To calculate simple interest:
Interest = principal rate time
I = Prt
The rate r, is expressed as a decimal when
calculating simple interest.
Compounding M

MATH 125: Quiz-01
Your Name:
1. Explain the idea of Percents.
2. Express
1
8
as a percent.
3. Express
1
4
as a decimal.
4. What you mean by Interest.
MATH 125: Quiz-01
Your Name:

MATH 125: Mathematics and Its Applications (3 hours)
1. Course Description: A diverse course including statistics and other topics such as mathematical
modeling, problem solving, finance, geometrical concepts, growth patterns, and applications to the
phys

Transition to Mathematical Proofs
Chapter 7 - Peano Arithmetic Assignment Solutions
Theorem 1 (Commutativity). For all a, b N, the following hold:
a+b=b+a
ab=ba
Theorem 2 (Associativity). For all a, b, c N, the following hold:
a + (b + c) = (a + b) + c
a