Computer Science Foundation Exam
December 18, 2009
Section II B
DISCRETE STRUCTURES SOLUTIONS
NO books, notes, or calculators may be used,
and you must work entirely on your own.
Name: _
PID: _
In this section of the exam, there are four (4) problems.
You
Mathematical Database
SETS AND FUNCTIONS
1. Introduction of Sets
The concept of sets is the foundation of modern mathematics and definitions of most
fundamental mathematics terms are based on sets. Sets are welldefined collection of objects. Here,
welld
Math 471
Group Work #2
SOLUTIONS
Fall 2011
Problem 1. How many strings of seven distinct letters of the alphabet contain
(a): at least one vowel? [Hint: How many contain no vowels?]
SOLUTION: There are 21 nonvowels in the alphabet, so we have P (21, 7) =
COT 3100
Slides 07/12
Topics
Exam Review
Assigned Reading: Chapter 1.14.4
HW#4 is due today!
(your recitation)
Midterm #2 (07/17 23:50 pm)
1 hour 50 mins
5 big problems (may have subproblems):





Set and Identities Proof (examples on the lectur
COT 3100
Slides 07/03
Assignment #3 (Due 07/05)
Physical Copy Only!
Turn in it to your TA during your recitation.
Check your graders Office Hours.
Explanations for HW#3 Problem 4.3 #35:
 xR in problem #35 represents the reversal of x.
 "The reversal of
COT 3100
Slides 07/05
Assignment #3 (Due 07/05
today!)
Physical Copy Only!
Turn in it to your TA during your recitation.
Check your graders Office Hours.
Explanations for HW#3 Problem 4.3 #35:
 xR in problem #35 represents the reversal of x.
 "The rever
COT 3100
Slides 06/02
Topics
Proofs
Assigned Reading: Chapter 1.6, 1.7
[Rosen] or Chapter 3.3 [Stein]
Sets
Assigned Reading: Chapter 2.1 [Rosen]
Announcements
HW#1 is due on 06/07 during your recitations!
Physical copy!
Unless you make other arrangements
COT 3100
Slides 07/24
Topics
Equivalence Relations
Assigned Reading: Chapter 8.5
Assignment #5
(Completed, Due 07/26)
Complete the following exercises from
chapter 8.5:
 14, 16, 42, 44, 46
Assignment #5 is due during your recitation on
Thursday 07/26. (N
COT 3100
Slides 06/28
Topics
Recursion
Structural Induction
No MTQ today
Assigned Reading: Chapter 4.3 [Rosen]
Assignment #3 (due 07/05)
Complete the following exercises from
chapter 4.3:
 2, 4, 8, 12, 22, 24, 34, 35, 48, 50 (you
can do all these problem
COT 3100
Slides 05/24
Assignment #1 (due 06/07)
You should start working on problems for 1.4
and 1.5 after this lecture (Based on [Rosen ]6th
Edition)
 Complete the following exercises from
chapter 1.4:
14, 28, 32(a,b, and d)
 Complete the following ex
COT 3100
Slides 09/06
Topics
Proofs
Assigned Reading: Chapter 1.6, 1.7
[Rosen] or Chapter 3.3 [Stein]
HW#1 (due 06/07)
You should start working on problems for 1.6
and 1.7 after this lecture (Based on 6th
Edition[Rosen])
 Complete the following exercises
COT 3100
Slides 06/26
Topics
Recursion
Structure Induction
Assigned Reading: Chapter 4.3
HW#3 is due on 07/05!
Complete the following exercises from
chapter 4.3:
 2, 4, 8, 12, 22, 24, 34, 35, 48, 50 (you
can do all these problems after this
lecture)
Recu
Discrete I
SAMPLE FINAL
March 3, 2008
NOTE: This is a sample final for MA2312. It is meant as a tool to review the topics
of this course. It should NOT be used as an indicator of length, difficulty, style, or
focus of the actual final. This sample final w
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CC T U9
&Ykl tC &1XCoA Gn
U t
9 b
e
e e CB Ai
pFWXA&UGGWvkzC SYz UC WipT Tz WSIDU` ` 7cfw_
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Math 55: Discrete Mathematics
UC Berkeley, Spring 2012
Homework # 6, due Wednesday, March 7
6.1.8 How many dierent threeletter initials with none of the letters repeated can people have?
There are 26 choices for the rst letter, 25 for the second, and 24
COT 3100Intro to Discrete Structures
Assignment 3
(From H. Rosens book 6th edition)
1. Complete the following exercises from chapter 4.1:
4.1.4 Let P (n) be the statement that 13 + 23 + + n3 = (n(n + 1)/2)2 for
the positive integer n.
a) What is the stat
COT 3100CIntro to Discrete Structures
Assignment 2
(From H. Rosens book 6th edition)
1. Complete the following exercises from chapter 2.1: (For 20 in particular,
justify your answer.)
2.1.8 Determine whether these statements are true or false.
a) cfw_
c)
COT 3100Intro to Discrete Structures
Assignment 4
(From H. Rosens book 6th edition)
1. Complete the following exercises from chapter 3.4:
3.4.10 What are the quotient and remainder when
a) 44 is divided by 8?
b) 777 is divided by 21?
c) 123 is divided b
Nongovernmental organisations come in many shapes and forms, with
differing goals and activities. Some concern themselves directly or indirectly
with policy; others do not. The concept of NGO is here applied specifically
to nonprofit organisations rathe