CS 1050: Constructing Proofs
Solutions to Problem Set 1
Problem 1
Direct Proofs (10 points)
1. Prove that for every real number x, x2 2x + 2 > 0.
Proof. x2 2x + 2 = (x 1)2 + 1
(x 1)2 0
(x 1)2 + 1 > 0
2. Suppose a and b are integers such that a b is divis
Cody Dansby
Decoupling Model Checking from Local-Area Networks in Semaphores
Cody Dansby and John Dorian
Abstract
Recent advances in real-time configurations and game-theoretic epistemologies offer a viable
alternative to the Ethernet. In fact, few end-us
RIGHT-PARTIAL FUNCTORS FOR AN ERDOS,
CONTINUOUSLY CO-INJECTIVE TOPOS
CODY DANSBY
Abstract. Let F . It was Euclid who rst asked whether freely
Gauss, trivially projective, super-almost separable homomorphisms can
be classied. We show that
0
(z) =
log (| G
Some Compactness Results for
LaplaceThompson Monodromies
Cody Dansby
Abstract
Let us suppose Volterras criterion applies. It has long been known
that
lim sup , u
c
(R) =
e dQ,
Q
[16]. We show that = t. Is it possible to derive left-smoothly Heavisid
Perfect Archetypes
Cody Dansby and John Dorian
Abstract
The rest of this paper is organized as follows. We
motivate the need for hash tables. We prove the deThe operating systems solution to spreadsheets is de- ployment of expert systems. Ultimately, we c
The Inuence of Probabilistic Methodologies on Networking
Thomer M. Gil
Abstract
tubes. However, the simulation of agents
might not be the panacea that information theorists expected. Nevertheless, this approach is
never adamantly opposed. We emphasize tha
An Improvement of the Location-Identity Split
Using GodSarn
Cody Dansby
A BSTRACT
Unied autonomous communication have led to many practical advances, including DHCP and Internet QoS. After years
of compelling research into e-business, we conrm the signica
Towards the Improvement of Context-Free Grammar
Cody Dansby
A BSTRACT
Computational biologists agree that relational congurations
are an interesting new topic in the eld of operating systems, and systems engineers concur. In fact, few electrical
engineers
The Impact of Permutable Technology on Algorithms
Cody Dansby
Abstract
technology. Despite the fact that conventional wisdom states that this grand challenge is generally addressed by the investigation of DHTs, we believe that a dierent solution is necess
Decoupling Smalltalk from Markov Models in Robots
Cody Dansby
Abstract
agents. Though similar applications emulate
interactive symmetries, we realize this intent
without enabling model checking.
We question the need for Smalltalk. we
view robotics as foll
CS 1050B: Constructing Proofs
Supplementary Exercises 3 : Comprehensive
Nov 29, 2006
Rosen Chapter 1 : Logic and Proofs
1. Chapter 1 Supplementary P.40
Assuming the truth of the theorem that states that is irrational whenever n is a positive
n
integer tha
CS 1050B: Constructing Proofs
Supplementary Exercises 1 : Proof, Induction, and Recursion
Answer Key
Problem 1 : Rosen Chapter 1 Supplementary Exercises
1. Problem 4
a) converse: If I drive to work today, then it will rain.
contrapositive: If I do not dri
CS 1050B: Constructing Proofs
Problem Set 2 : Induction and Recursion
Due Friday, Sept 22nd, after the class
Problem 1 : Rosen 4.1: 3, 4, 17
1. Let P (n) be the statement that 12 + 22 + + n2 =
n(n+1)(2n+1)
6
for the positive integer n.
a) Plugging in n =
CS 1050B: Constructing Proofs
Problem Set 3 :
Due Monday, Oct 9th, after the class
1. Problem 1 : Rosen 7.1: 8
Find the solution to each of these recurrence relations with the given initial conditions. Use
an iterative approach such as that used in Exampl
CS 1050B: Constructing Proofs
Problem Set 4 :
Due Wednesday, Oct 18th, after the class
1. Rosen 5.1: 22
How many positive integers between 1000 and 9999 inclusive
Answer:
It will be useful to note rst that there are exactly 9000 numbers in this range
a) a
CS 1050B: Constructing Proofs
Problem Set 5
Due Wednesday, Oct 25th, after the class
1. Rosen 5.3: 30
Seven women and nine men are on the faculty in the mathematics department at a school
a) How many ways are there to select a committee of ve members of t
CS 1050B: Constructing Proofs
Problem Set 6
Due Wednesday, Nov 3rd, after the class
1. Rosen 6.1: 36
Rolling dice:
a) What is the total number of possible outcomes of the roll of two dice? and three dice?
b) How many ways are there to get a total of 8 whe
CS 1050B: Constructing Proofs
Problem Set 7
Due Friday, Nov 20th, after the class
1. Rosen 3.4: 16
Evaluate these quantities.
a) -17 mod 2
b) 144 mod 7
c) -101 mod 13
d) 199 mod 19
Answer:
a) Since 17 = 2 (9) + 1, the remainder is 1. That is, 17 mod 2 = 1
CS 1050B: Constructing Proofs
Problem Set 8
Due Wednesday, Nov 29th, after the class
1. Rosen 3.7: 4 (c)(d)(f ) only
Express the greatest common devisors of each of these pairs of integers as a linear combination
of these integers.
a) 35, 78
b) 21, 55
c)
CS 1050B: Constructing Proofs
Quiz 1
1. (10 points) Prove that
2 is irrational by giving a proof by contradiction.
The proof can be found on page 80 of Rozen. It is also closely related to the proof of problem
2.2 in Homework 2.
2. The factorial n! is den
CS 1050B: Constructing Proofs
Quiz 2
1. (10 points) Find the solution to each of these recurrence relations and initial conditions.
a) an = an1 + n, a0 = 1
b) an = 2nan1 , a0 = 1
Solution :
a)
an = n + an1
= n + (n 1) + an2 )
= (n + (n 1) + (n 2) + an3 )
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