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Discrete Mathematics

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Submitted By justinellis112
Words 1156
Pages 5
* -------------------------------------------------
Homework problems:
Section 1.1, pages 12–16: #2, #11, #31
Section 1.2, pages 22–24: #3, #8a,b,c, #24
Section 1.4, pages 53–55: #6, #11, #32
Section 2.3, pages 152–153: #2, #12, #13
Section 2.6, pages 183–184: #2a, #4b * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
2. Which of these are propositions?What are the truth values * ------------------------------------------------- of those that are propositions? * -------------------------------------------------
a) Do not pass go. -No * -------------------------------------------------
b) What time is it? - No * -------------------------------------------------
c) There are no black flies in Maine. - Yes, FALSE * -------------------------------------------------
d) 4 + x = 5. - No * -------------------------------------------------
e) The moon is made of green cheese. Yes, FALSE * ------------------------------------------------- f ) 2n ≥ 100. , No * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
11. Let p and q be the propositions * ------------------------------------------------- p : It is below freezing. * ------------------------------------------------- q : It is snowing. * -------------------------------------------------
Write these propositions using p and q and logical connectives * -------------------------------------------------
(including negations). * -------------------------------------------------
a) It is below freezing and snowing. p ^ q * -------------------------------------------------
b) It is below freezing but not snowing. p ^ ~q * -------------------------------------------------
c) It is not below freezing and it is not snowing. ~p^~q * -------------------------------------------------
d) It is either snowing or below freezing (or both). (p v q) * -------------------------------------------------
e) If it is below freezing, it is also snowing. p → q * ------------------------------------------------- f ) Either it is below freezing or it is snowing, but it is * ------------------------------------------------- not snowing if it is below freezing. (p v q) ^ (p → ~q) * -------------------------------------------------
g) That it is below freezing is necessary and sufficient * ------------------------------------------------- for it to be snowing. p ← → q * -------------------------------------------------

* -------------------------------------------------
31. Construct a truth table for each of these compound propositions. (a-f) * -------------------------------------------------
a) p ∧¬p * -------------------------------------------------

* ------------------------------------------------- p | * ------------------------------------------------- p ∧¬p | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * ------------------------------------------------- * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
b)
* -------------------------------------------------
) p ∨¬p * ------------------------------------------------- p | * ------------------------------------------------- p v ~p | * ------------------------------------------------- t | * ------------------------------------------------- t | * ------------------------------------------------- f | * ------------------------------------------------- t | * -------------------------------------------------

* -------------------------------------------------
c) (p ∨¬q) → q * -------------------------------------------------

* ------------------------------------------------- p | * ------------------------------------------------- q | * -------------------------------------------------
(p ∨¬q) → q | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* ------------------------------------------------- d) (p ∨ q) → (p ∧ q) * ------------------------------------------------- p | * ------------------------------------------------- q | * -------------------------------------------------
(p ∨ q) → (p ∧ q) | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
e) (p → q) ↔ (¬q →¬p) * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* ------------------------------------------------- p | * ------------------------------------------------- q | * -------------------------------------------------
(p → q) ↔ (¬q →¬p) | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* ------------------------------------------------- f ) (p → q) → (q → p) * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* ------------------------------------------------- p | * ------------------------------------------------- q | * -------------------------------------------------
(p → q) → (q → p) | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
F | * -------------------------------------------------
T | * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
*In Exercises 1–6, translate the given statement into propositional * ------------------------------------------------- logic using the propositions provided. * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
3. You can graduate only if you have completed the requirements * ------------------------------------------------- of your major and you do not owe money to the * ------------------------------------------------- university and you do not have an overdue library book. * -------------------------------------------------
Express your answer in terms of g: “You can graduate,” * ------------------------------------------------- m: “You owe money to the university,” r: “You have completed * ------------------------------------------------- the requirements of your major,” and b: “You have * ------------------------------------------------- an overdue library book.” * -------------------------------------------------
(r ^ ~m ^ ~b) → g * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
8. Express these system specifications using the propositions * ------------------------------------------------- p “The user enters a valid password,” q “Access is * ------------------------------------------------- granted,” and r “The user has paid the subscription fee” * ------------------------------------------------- and logical connectives (including negations). * -------------------------------------------------

* -------------------------------------------------
a) “The user has paid the subscription fee, but does not * ------------------------------------------------- enter a valid password.” * -------------------------------------------------

* -------------------------------------------------
(r ^ ~p) * -------------------------------------------------
b) “Access is granted whenever the user has paid the * ------------------------------------------------- subscription fee and enters a valid password.” * -------------------------------------------------

* -------------------------------------------------
(r^p) → q * -------------------------------------------------
c) “Access is denied if the user has not paid the subscription * ------------------------------------------------- fee.” * -------------------------------------------------

* -------------------------------------------------
~r → ~p * -------------------------------------------------

* -------------------------------------------------
Exercises 24–31 relate to inhabitants of an island on which * ------------------------------------------------- there are three kinds of people: knights who always tell the * ------------------------------------------------- truth, knaves who always lie, and spies (called normals by * -------------------------------------------------
Smullyan [Sm78]) who can either lie or tell the truth. You * ------------------------------------------------- encounter three people, A, B, and C. You know one of these * ------------------------------------------------- people is a knight, one is a knave, and one is a spy. Each of the * ------------------------------------------------- three people knows the type of person each of other two is. For * ------------------------------------------------- each of these situations, if possible, determine whether there * ------------------------------------------------- is a unique solution and determine who the knave, knight, and * ------------------------------------------------- spy are. When there is no unique solution, list all possible * ------------------------------------------------- solutions or state that there are no solutions. * -------------------------------------------------

* -------------------------------------------------
24. A says “C is the knave,” B says, “A is the knight,” and C * ------------------------------------------------- says “I am the spy.” * -------------------------------------------------

* -------------------------------------------------
A is the knight, Bis the spy, C is the knave * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
6. Let N(x) be the statement “x has visited North Dakota,” * ------------------------------------------------- where the domain consists of the students in your school. * -------------------------------------------------
Express each of these quantifications in English. * -------------------------------------------------
a) ∃xN(x) * ------------------------------------------------- there is a student at my school that has visited North Dakota * -------------------------------------------------
) b) ∀xN(x) * ------------------------------------------------- every student at my school has visited North Dakota * -------------------------------------------------
c) ¬∃xN(x) * ------------------------------------------------- there is not a student at my school that has visited North Dakota * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
d) ∃x¬N(x) * ------------------------------------------------- there is a student at my school that has not gone to North Dakota * ------------------------------------------------- e) ¬∀xN(x) * -------------------------------------------------
There is someone at my school that has not gone to North Dakota * ------------------------------------------------- f ) ∀x¬N(x) * ------------------------------------------------- every student at my school has not gone to North Dakota * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
11. Let P(x) be the statement “x = x2.” If the domain consists * ------------------------------------------------- of the integers, what are these truth values? * -------------------------------------------------
a) P(0) true * ------------------------------------------------- b) P(1) true * ------------------------------------------------- c) P(2) false * -------------------------------------------------
d) P(−1) * -------------------------------------------------

* ------------------------------------------------- true * -------------------------------------------------

* ------------------------------------------------- e) ∃xP(x) * ------------------------------------------------- true * ------------------------------------------------- f) ∀xP(x) * ------------------------------------------------- false * -------------------------------------------------

* -------------------------------------------------
32. Express each of these statements using quantifiers. Then * ------------------------------------------------- form the negation of the statement so that no negation is * ------------------------------------------------- to the left of a quantifier. Next, express the negation in * ------------------------------------------------- simple English. (Do not simply use the phrase “It is not * ------------------------------------------------- the case that.”) * -------------------------------------------------
a) All dogs have fleas. ∀ dF(d), ∃-F(d), there is a dog that does not have fleas. * -------------------------------------------------
b) There is a horse that can add.∃hA(h),∀-A(h), All horses cannot add. * -------------------------------------------------
c) Every koala can climb.∀ kC(k),∃k-C(k), there is a koala that cannot climb. * -------------------------------------------------
d) No monkey can speak French. -∃mF(m), ∃mF(m), there is a monkey that can speak french. * -------------------------------------------------
e) There exists a pig that can swim and catch fish. ∃p(S(p)^F(p)). , ∀p(-S(p)^-F(p)), * ------------------------------------------------- all pigs can neither swim nor catch fish. * -------------------------------------------------

* -------------------------------------------------
2. Determine whether f is a function from Z to R if * -------------------------------------------------
a) f (n) = ±n.no * -------------------------------------------------
b) f (n) = √n2 + 1.yes * -------------------------------------------------
c) f (n) = 1/(n2 − 4). yes * -------------------------------------------------

* -------------------------------------------------
12. Determine whether each of these functions from Z to Z * ------------------------------------------------- is one-to-one. * -------------------------------------------------
a) f (n) = n − 1 , yes b) f (n) = n2 + 1, no * -------------------------------------------------
c) f (n) = n3 yes d) f (n) = n/2, yes * -------------------------------------------------

* -------------------------------------------------
13. Which functions in Exercise 12 are onto? * -------------------------------------------------

* -------------------------------------------------
a)yes, b)no, c)no, d)no * -------------------------------------------------

* -------------------------------------------------
2. Find A + B, where * -------------------------------------------------

* -------------------------------------------------
A= [ 1 0 4] * ------------------------------------------------- [ -1 2 2] ***Boolean Matrix * ------------------------------------------------- [0 -2 -3] * -------------------------------------------------

* -------------------------------------------------
B= [ -1 3 5 ] * ------------------------------------------------- [ 2 2 -3] * ------------------------------------------------- [2 -3 0] * -------------------------------------------------
A+B = * -------------------------------------------------
0 | * -------------------------------------------------
3 | * -------------------------------------------------
9 | * -------------------------------------------------
1 | * -------------------------------------------------
4 | * -------------------------------------------------
-1 | * -------------------------------------------------
2 | * -------------------------------------------------
-5 | * -------------------------------------------------
-3 | * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
4. Find the product of AB where: * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
A = [1 -3 0 ] , B = [ 1 -1 2 3 ] * ------------------------------------------------- [1 2 2 ] [-1 0 3 -1 ] * ------------------------------------------------- [ 2 1 -1] [-3 -2 0 2 ] * -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------

* -------------------------------------------------
4 | * -------------------------------------------------
-1 | * -------------------------------------------------
-7 | * -------------------------------------------------
6 | * -------------------------------------------------
-7 | * -------------------------------------------------
-5 | * -------------------------------------------------
8 | * -------------------------------------------------
5 | * -------------------------------------------------
4 | * -------------------------------------------------
0 | * -------------------------------------------------
7 | * -------------------------------------------------
3 | * -------------------------------------------------

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...1. Current scenario As an established auto parts shop, the company are looking forward to extend their business reach to its customers and also further increase their reputation in the automotive industry. As of now, the company only provides their services within their shop. Customers can either call or come to their shop to order and buy or make an inquiries regarding a specific products or parts. Most of the customers are the local resident of Brunei and are yet to cater customers from other countries. In this era of e-commerce, the company are well aware of the advantages of setting up an online auto shop to gain a competitive advantage over its competitor in the industry as there are currently no local auto parts shop have set up an online auto shop. Customers can browse and buy specific parts with detailed information on the parts according to the customer’s car model and year. This makes it easier and less time consuming for the customers. Setting up an online auto shop also enables the company to cater customers from other countries and enter the international markets. In terms of marketing, the company carry out their advertisement through newspapers, brochures and banners. This methods of advertisement does work for the company however with the increase use of smart phones, tablets and computers, going online will help them more to reach out to more customers locally and internationally where they can provide news, updates on their new products and services...

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Signal and System

...Signal may be either continuous-time or discrete-time, with either analog or digital values [1]. The signals which are represented by a continuous function are called continuous signals and those which are described by number sequences are called discrete signals [2]. We have seen about a signal in brief. The second component in signal processing is a system which is a process whose input and output are signals. Signal processing is a vast area comprising the concepts of electrical engineering, systems engineering and applied mathematics that deals with both the analog and discrete time signals, represented by variation in time or spatial physical quantities. Precise statistical depiction is required for the development of improved signal processing algorithms of natural signals [3]. The major operations of Signal processing includes 1) signal acquisition and reconstruction, 2) Quality improvement including filtering, smoothing and digitization, 3) feature extraction 4) signal compression and 5) prediction [4] [5]. Analog signal processing, Discrete-time signal processing, Non-linear signal processing and Digital signal processing are the four major categories of signal processing. The signal processing performed over analog signals for the purpose of any of the major operations of signal processing is known to be analog signal processing and the same concept is applied for discrete-time signal processing, where the only difference is discrete signal is employed. An analog signal...

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Mth221Syllabus

... | | |Discrete Math for Information Technology | Copyright © 2010 by University of Phoenix. All rights reserved. Course Description Discrete (as opposed to continuous) mathematics is of direct importance to the fields of Computer Science and Information Technology. This branch of mathematics includes studying areas such as set theory, logic, relations, graph theory, and analysis of algorithms. This course is intended to provide students with an understanding of these areas and their use in the field of Information Technology. Policies Faculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: • University policies: You must be logged into the student website to view this document. • Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies at the beginning of each class. Policies may be slightly different depending on the modality in which you attend class. If you have recently changed modalities, read the policies governing your current class modality. Course Materials Grimaldi, R. P. (2004). Discrete and combinatorial mathematics: An applied introduction. (5th ed.). Boston, MA: Pearson Addison Wesley. Article...

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