SHEN'S CLASS NOTES-191
Chapter Two The Language of Mathematics
In this chapter, we discuss sets, functions, and sequences. These are very fundamental notions or language to model mathematical problems as well as tools for solving mathematical proble
F06/CS 191 :
Suggested Solutions to TEST TWO
Discrete Structures I OPEN BOOK
There are 5 problems with 20 points each. 1
11/2/2006 Name_
Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Define the relation R on X as follows. (a, b) R
Solutions to Homework Four CS 191 : Discrete Structures I
Mar. 6, Winter 2007 Due Date: Mar. 20, 2007, Tuesday, before the class meeting.
Section 4.2, p155
Exercise (15) Write an algorithm whose input is a sequence s1, ., sn sorted in nondecreasing
CS191 Homework One
Winter 2007 Due: Jan. 30, 2007, in class
Section 1.1, p7
Determine whether each sentence in Exercises 1-8 is a proposition. If the sentence is a proposition, write its negation. (you are not being asked for the truth values of the
Solutions to Homework Five CS 191 : Discrete Structures I
Apr. 3, Winter 2007 Due Date: Apr. 17, 2007, Tuesday, before the class meeting.
Section 5.2, p205
In Exercise 26-31, express each hexadecimal number in decimal. Exercise (30) 209D Answer: (20
Solutions to Homework Six CS 191 : Discrete Structures I
Apr. 17, Winter 2007 Due Date: Apr. 26, 2007, Thursday, before the class meeting.
Section 6.6, p265
In Exercise 1-3, determine the number of strings that can be formed by ordering the letters
F06/CS 191 :
Suggested Solutions to TEST ONE
Discrete Structures I OPEN BOOK
There are 5 problems with 20 points each.
9/28/2006 Name_
1. Use truth tables to determine whether (p logically equivalent or not. Answer : p T T T T F F F F q T T F F T
Ch. 2.1 2.2: Proofs
Kenneth Mitchell
School of Computing & Engineering,
University of Missouri-Kansas City,
Kansas City, MO 64110
Kenneth Mitchell, CS & EE dept., SCE, UMKC p. 1/14
Mathematical Systems
A mathematical system consists of:
1. Axioms Statemen
Ch. 1.2 1.4: Logic
Kenneth Mitchell
School of Computing & Engineering,
University of Missouri-Kansas City,
Kansas City, MO 64110
Kenneth Mitchell, CS & EE dept., SCE, UMKC p. 1/21
Introduction
Which sentences are true
1. The only positive integers that di
Ch. 6.1: Counting
Kenneth Mitchell
School of Computing & Engineering,
University of Missouri-Kansas City,
Kansas City, MO 64110
Kenneth Mitchell, CS & EE dept., SCE, UMKC p. 1/17
Counting
Counting is useful in analyzing algorithms and essential to probabi
Ch. 2.4: Mathematical Induction
Kenneth Mitchell
School of Computing & Engineering,
University of Missouri-Kansas City,
Kansas City, MO 64110
Kenneth Mitchell, CS & EE dept., SCE, UMKC p. 1/13
Introduction
Suppose we have an innite long sequence of blocks
Ch. 6.8: Pigeonhole Principle
Kenneth Mitchell
School of Computing & Engineering,
University of Missouri-Kansas City,
Kansas City, MO 64110
Kenneth Mitchell, CS & EE dept., SCE, UMKC p. 1/6
Pigeonhole principle
First form:
If n pigeons fly into k pigeonho
CS291: Quiz1 Fall 2010 Oct 12
NAME:
This exam has six questions (for 120 points). You are to write legibly, show all work. I cannot
give partial credit if no work is shown. References are made to definitions and methods as used
in class and/or book. If yo
CS291: Quiz2 April 13 2010
NAME:
Degree program
This exam has five questions (for 100 points). You are to write legibly, show all work. I cannot
give partial credit if no work is shown. References are made to definitions and methods as used
in class and/o
CS291: Quiz2 Winter 2011 April 11
NAME:
Degree program
This exam has five questions (for 150 points). You are to write legibly, show all work. I cannot
give partial credit if no work is shown. References are made to definitions and methods as used
in clas
CS291: Quiz1 Review 2010
1. For the following graph tell whether the given path is
a) A simple path
b) A cycle
c) A simple cycle
d) None of the above
e9
A
B
e4
C
e2
e1
e3
e5
e6
D
E
e8
F
e7
(a) (A,e2,E,e3,B,e4,C,e5,F,e8,E,e6,D,e1,A).
(b) (B,e3,E,e3,B,e4,C,
SHEN'S CLASS NOTES-191
Class Notes for Discrete Structures I CS 191 Jan. 10, 2005 Textbook:
Discrete Mathematics
6/E
By Richard Johnsonbaugh
1
SHEN'S CLASS NOTES-191
Chapter One Logic and Proofs
Logic is the study of reasoning. It provides reas
Homework Four CS 191 : Discrete Structures I
Mar. 6, Winter 2007 Due Date: Mar. 20, 2007, Tuesday, before the class meeting.
Section 4.2, p155
_ Exercise (15) Write an algorithm whose input is a sequence s1, ., sn sorted in nondecreasing order and a
SHEN'S CLASS NOTES-191
Chapter Three Relations
3.1 Relations Relations generalize the notion of functions. Definition 3.1 A (binary) relation R from a set X to a set Y is a subset of the Cartesian product X Y. If (x, y) R, we say that x is related
SHEN'S CLASS NOTES-191
Chapter Four Algorithms
4.1 Introduction An algorithm is a well-defined computational procedure to solve a computational problem (or problem for short). An algorithm must be correct. A good algorithm must be readable, time-eff
F06/CS 191 :
Suggested Solutions to TEST THREE
Discrete Structures I OPEN BOOK
12/12/2006, 5:45-7:45pm Name_
There are 5 problems with 20 points eachplus a makeup problem for the second test.
1 (a)
Convert the following decimal number into a he
SHEN'S CLASS NOTES-191
Chapter Five Introduction to Number Theory
Number Theory is the branch of mathematics concerned with the integers.
5.1 Divisors
Definition 5.1.1 Let n and d be two integers, d 0. We say that d divides n if there exists an i
SHEN'S CLASS NOTES-191
Chapter Six Counting Methods and The Pigeonhole Principle
6.1 Basic Principles
Example 1 Suppose Kay's Quick Lunch has the following lunch menu.
APPETIZERS Nachos . 2.15 Salad . 1.90 MAIN COURSES Hamburger . 3.15 Cheeseburge
Homework Two CS 191 : Discrete Structures I
Jan. 30, Winter 2007 Due Date: Feb. 13, Tuesday, before the class meeting.
Section 1.7, p60
In Exercises 1-11, using induction, verify that each equation is true for every positive integer n. Exercise (6)
Homework Three CS 191 : Discrete Structures I
Feb. 20, Winter 2007 Due Date: Mar. 6, Tuesday, before the class meeting.
Section 3.1, p123
In Exercises 9-12, draw the diagraph of the relation. Exercise (11) The relation R = {(1,2), (2,3), (3,4), (4,1
W07/CS 191 :
Suggested Solutions to TEST ONE
Discrete Structures I OPEN BOOK
There are 5 problems with 20 points each.
2/15/2007 Name_
1. Use truth tables to determine whether (p equivalent or not. Answer : p T T T T F F F F q T T F F T T F F r T
Solutions to CS191 Homework One
Winter 2007 Due: Jan. 30, 2007, in class
Section 1.1, p7
Determine whether each sentence in Exercises 1-8 is a proposition. If the sentence is a proposition, write its negation. (you are not being asked for the truth
Solutions to Homework Two CS 191 : Discrete Structures I
Jan. 30, Winter 2007 Due Date: Feb. 13, Tuesday, before the class meeting.
Section 1.7, p60
In Exercises 1-11, using induction, verify that each equation is true for every positive integer n.
Solutions to Homework Three CS 191 : Discrete Structures I
Feb. 20, Winter 2007 Due Date: Mar. 6, Tuesday, before the class meeting.
Section 3.1, p123
In Exercises 9-12, draw the diagraph of the relation. Exercise (11) The relation R = {(1,2), (2,3)