Doerr
Week 5
An Introduction to Logic, Propositions & Logical Operators
I repeat here several of the tables you studied in week 1. In week 1 we reminded ourselves of a
few basic properties/laws in algebra. Our study of sets and basic set laws convinced us
Exam I
Discrete Structures I
Directions: Exam 1 is due Monday, September 29th by 8am. Answer all questions carefully.
Do all steps with brief explanations similar to what was done in the examples in the notes. You
can scan your exam and send it as a pdf f
92.221
PROJECT ( due at the time and date of exam 1)
You may work together, but submit one paper with the names of all who worked on
the project . Each person must contribute equally. This project is worth 10 points
toward exam I. Up to 5 additional point
A. Doerr
Week 4
Laws of Addition, Combinations, the Binomial Coefficient
Laws of Addition
Example 1. A student must select a project from one of three lists provided by the
professor There are 5 projects on the first list, 10 on the second and 12 on the t
A. Doerr
Week 3 Basic Counting Techniques The Rule of Products
WHAT IS COMBINATORICS?
One of the first concepts our parents taught us was the "art of counting." We were taught
to raise three fingers to indicate that we were three years old. The question o
Week 12
An Introduction to the Concept of Relations
For weeks 12 and 13 we will be following the text closely. Here is some additional material.
Think of your family tree. It consists of your brothers, sisters, cousins, second cousins, aunts,
uncles, pare
A. Doerr
Week 9
Minsets, The Duality Principle and Basic Matrix Definitions
We will cover this material as it is given in the text and in class.
Before reading the text study the following list of basic matrix algebra laws.
Some Basic Matrix Laws
Assume t
A. Doerr
Week 11
Solving Systems of Equations Using Matrices and Network Analysis
This topic is quite lengthy and has several parts. I consider this extra material which we
will cover if we have time. The examples in Part 3 are quite interesting and the m
92.321
PRACTICE EXAM II
Check week 6 notes
1. Use mathematical induction to prove:
1
for every positive integer n.
n 1
2. (a) Let n be a positive integer. Use the proof by contradiction method of proof to prove: (explain the
procedure and all steps). If 5
Doerr
Week 6
Equivalence, Implication and the Laws of Logic
Equivalences and Implications
In section 3.3 there are several key terms; tautology, contradiction, equivalence and
implication. Study the definitions of these terms before reading further.
Examp
Doerr
Week 8
Methods of Proving Theorems
Why?
Almost nothing in undergraduate, and some graduate, mathematics courses creates FEAR
more than when an instructor announces that the students will be required to do proofs. Yet, we
all prove statements frequen
Doerr
Week 7
Before we discuss math induction a brief comment. Example 3.7.5 define prime number.
A famous conjecture (what does this term mean) is Goldbach's conjecturewhich is one of the
oldest and best-known unsolved problems in number theory and in al
Exam I
Discrete Structures
Directions: The exam is due no later than Wednesday June 8th at 8am. Late exams
will not be accepted.
Answer all questions carefully. Do all steps with detailed explanations. I prefer that
you send your solutions electronically.
Doerr
Week 14
Closure of Relations and an Introduction to Functions
Composition of Relations Revisited
Recall in the text the matrix of the relation S R is written as MS R and
MS R = MR MS. The purpose of the following example is to remind you of the proc
Doerr
Copyright 1996
(Previously in 1986 and 1994). These notes are for the exclusive use
of Prof. Alan Doerrs classes and may not be used and/or reproduced in any form or by
any means, without permission in writing from Prof. Alan Doerr.
Week 1: Introdu
Doerr
Week 2
More on Sets and Summation
I will comment on the topics of week 1 given in the text. Read the text and then read the
following.
Sets:
You can think of week 1 as a combination of learning notation and the structure of the
mathematical system,
92.321
Practice Exam I
Directions: A main focus of this course is that each person develop the logical
processes necessary to explain/prove their results in a clear, precise and logical
fashion using relevant definitions. Your grade will base on your expl
Theorems on invertible Matrices
Let A be an n x n matrix. Let V and W be vector spaces over R, and let T : V
W be
a linear transformation whose matrix is A. (with respect to some bases of V and W).
Then the following statements are equivalent:
a) A is inv
92.321
Practice Exam I
Here are a few additional practice problems.
You are also responsible for all homework problems and examples covered in class. Ask
questions on the exam in class anytime.
1.
(a) Use truth tables to determine whether the following is
92.321
PRACTICE EXAM 3
1 1 0 3
1. The following matrix represents a system of equations. 1 1 1 1
0 1 1 2
(a) Write this system of equations
(b) Solve this system (in matrix notation) using the Gauss Jordan technique
2. Solve the system of equation rep
Properties and some facts about determinants
Let A = [aij] be a n x n matrix with real or complex entries.
1. If A has a 0 row (or column) then detA = 0.
2. If a matrix B is obtained from the matrix A by interchanging any two rows (or columns)
then the de
The data is for weekly sales in the dry goods department at a Wal*Mart store in the
Northeast. Peak values, I.e. spikes, usually occur at holiday periods. Week 1 is the
first week of February 2003. To show continuity, week 1 of 2004 is represented as
week
Chan 1
Pengtaingvouch Hok
BUS 110-51
James Dottin
December 3, 2014
MODULE 11
Exercise
1.
Who uses accounting information? What do they use it for, and why do they find it
helpful? What problems would arise if they werent provided with accounting informati
Certicate
of completion
Pengtaingvouch Hok
has successfully completed the HP LIFE e-Learning
course on Hiring sta
Through this self-paced online course, totaling approximately 1 Contact Hour, the above
participant actively engaged in an exploration of the
Certicate
of completion
Pengtaingvouch Hok
has successfully completed the HP LIFE e-Learning
course on Cash ow
Through this self-paced online course, totaling approximately 1 Contact Hour, the above
participant actively engaged in an exploration of how to
Certicate
of completion
Pengtaingvouch Hok
has successfully completed the HP LIFE e-Learning
course on Eective leadership
Through this self-paced, short online course, totaling approximately 1 Contact Hour, the above
participant actively engaged in an exp
92.321 Discrete Structures I
Instructor: Alan Doerr
Contact Information:
OFFICE: OH 428 U
TEL. 978-934-2415
email: Alan_Doerr@uml.edu
OFFICE HOURS: MWF 10:30 - 11:30, M from 1:00- 2:00 and W from 1:30 to 4:00 on
most days. Im on campus every day, most tim
A. Doerr
Week 4
Laws of Addition, Combinations, the Binomial Coefficient
Laws of Addition
Example 1. A student must select a project from one of three lists provided by the
professor There are 5 projects on the first list, 10 on the second and 12 on the t