CS 173: Discrete Structures, Fall 2010 Homework 8
This homework contains 3 problems worth a total of 38 regular points. It is due on Friday, October 29 at 4pm. Put your homework in the appropriate dropbox in the Siebel basement. 1. More on recurrences [12
Building Blocks for Theoretical Computer Science
Margaret M. Fleck
Chapter 1: Math Review
1.1 Some Sets
Z = cfw_. . . , 3, 2, 1, 0, 1, 2, 3, . . . is the integers.
N = cfw_0, 1, 2, 3, . . . is the non-negative integers, also known as the natural numbers
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AnExampleofHowaComputerReallyWorks Acomputerisacomplexsystemconsistingofmanydifferentcomponents.Butattheheart or the brain, if you want of the computer is a single component that does the actual computing.ThisistheCentralProcessingUnit,orCPU.Inamoderndesk
Question 1: Multiple Choice & Short Answer A. You are a software developer that is developing an inventory system for a store. The store currently stocks 9000 items and you would like to refer to them by a unique ID number. What is the minimum number of b
Reference Sheet You may separate this sheet from the exam, but please hand it in with your exam. Please do not include work on this page.
Boolean Algebra
x+0=x x+1=1 x+x=x x + x = 1 (x) = x x+y=y+x x + (y + z) = (x + y) + z x(y + z) = xy + xz (x + y) = xy
CS 173: Discrete Structures, Fall 2010 Quiz 2 Solutions
1. (1 point) Give the day and time when your assigned discussion section meets. State explicitly if you have switched sections recently. Solution: The right answer varies from person to person. 2. (4
CS 173: Discrete Structures, Fall 2010 Homework 1 Solutions
This homework contains two problems worth a total of 20 points, plus an on-line survey to complete. It is due on Friday, September 3rd at 4pm. The homework dropboxes are towards the east end of t
CS 173: Discrete Structures, Fall 2010 Homework 2 Solutions
This homework contains 5 problems worth a total of 47 points.
1. [9 points] Translating notation into English Suppose we dene: F (x) is x plays Fate. B (x) is x plays banjo. K (x) is x likes kimc
CS 173: Discrete Structures, Fall 2010 Homework 3 Solutions
This homework contains 4 problems worth a total of 38 points. It is due on Friday, September 17th at 4pm. When a problem species a particular proof technique, you must use that technique in your
CS 173: Discrete Structures, Fall 2010 Homework 4 Solutions
This homework contains 6 problems worth a total of 47 points. It is due on Friday, September 24th at 4pm.
1. Set Operations [12 points] Suppose you were given the following sets: A B C D E = = =
CS 173: Discrete Structures, Fall 2010 Homework 5 Solutions
This homework contains 5 problems worth a total of 50 points. It is due on Friday, October 8th at 4pm. 1. Functions [8 points] For each of the following functions, state what its image is, whethe
CS 173: Discrete Structures, Fall 2010 Homework 6 Solutions
This homework was worth a total of 54 points.
1. Recursive denition [13 points] Give a simple closed-form denition for each of the following subsets of the real plane. Give both a precise denitio
CS 173: Discrete Structures, Fall 2010 Homework 7
This homework contains 4 problems worth a total of 40 points. It is due on Friday, October 22 at 4:00 PM. Put your homework in the appropriate dropbox in the Siebel basement. 1. Proofs using Big-O and [9 p
CS 173: Discrete Structures, Fall 2010 Homework 8 Solutions
This homework contains 3 problems worth a total of 37 regular points and one bonus point.
1. More on recurrences [12 points] (a) (6 points) Derive closed form solutions for the following two recu
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Functions
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Wednesday, September 21, 2016
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Recursive Denitions
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1
CS 173, Fall 2016
NETID:
Examlet 10, Part B
FIRST:
LAST:
Discussion:
Thursday
2
3
4
5
Friday 9
10
11
12
1
2
1. (9 points) Fill in key facts about the recursion tree for T , assuming that n is even.
T (8) = 5
(a) The height:
T (n) = 3T (n 2) + c
n
2
4
(b
CS 173, Fall 2016
Examlet 8, Part B
NETID:
FIRST:
Discussion:
LAST:
Thursday
2
3
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(10 points) Suppose we have a function F defined (for n a power of 2) by
F (2) = c
F (n) = F (n/2) + n for n 4
Your partner has already figured out
1
CS 173, Fall 2016
Examlet 7, Part B
NETID:
FIRST:
LAST:
Discussion:
Thursday
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1. (9 points) What is the chromatic number of graph G (below)? Justify your answer.
t
B
D
C
R
R
s
G
y
x
p
q
G
B
n
G
B
B
A
B
R
R
m
G
Solution: The
1
CS 173, Fall 2016
Examlet 6, Part A
NETID:
FIRST:
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Discussion:
Thursday
2
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1. (10 points) How many isomorphisms are there from G (below) to itself? Justify your answer
and/or show your work clearly .
B
J
A
C
E
D
I
G
K
H
S
1
CS 173, Fall 2016
Examlet 8, Part A
NETID:
FIRST:
Discussion:
LAST:
Thursday
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(20 points) Let function f : N Z be defined by
f (0) = 2
f (1) = 7
f (n) = f (n 1) + 2f (n 2), for n 2
Use (strong) induction to prove that f (n)
1
CS 173, Fall 2016
Examlet 7, Part A
NETID:
FIRST:
Discussion:
LAST:
Thursday
2
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1
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Use (strong) induction to prove the following claim:
For any natural number n, 2n3 + 3n2 + n is divisible by 6.
Solution: Proof by induction on n.