MTH 221 MTH 221 MTH 221.doc

MTH 221 Week 1 Connect Exercises




MTH 221 Week 5 Final Exam

MTH 221 Week 5 Case Study

MTH 221 Entire Course

MTH 221 Week 1 Connect Exercises1. A particular brand of shirt comes in 13 colors, has a male version and a female version, and comes in 3 sizes for each sex. How many different types of this shirt are made?

2. How many strings of five decimal digits1. do not contain the same digit twice? 2. end with an even digit? 3. have exactly four digits that are 9s? 3. How many strings of six uppercase English letters are there1. if letters can be repeated?

2. if no letter can be repeated?

3. that start with X, if letters can be repeated?

4. that start with X, if no letter can be repeated?

5. that start and end with X, if letters can be repeated?

6. that start with the letters NE (in that order). if letters can be repeated?

7. that start and end with the letters NE (in that order), if letters can be repeated?

8. that start or end with the letters NE (in that order), if letters can be repeated?

4. In how many different orders can five runners finish a race if no ties are allowed?5. How many bit strings of length 9 have

exactly three O s?

more O s than 1 s?

at least six 1 s?

at least three 1 s?

6. A club has 16 members

a) How many ways are there to choose four members of the club to serve on an executive committee?

b) How many ways are there to choose a president. vice president. secretary. and treasurer of the club, where no person can hold more than one office?

7. Five women and nine men are on the faculty in the mathematics department at a schoola) How many ways are there to select a committee of five members of the department if at least one woman must be on the committee?

b) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee?

8. Find the coefficient of in x16y4 in (x + y)20

9. What is the coefficient of x8 in (3 + x)12 ?

10. In how many different ways can seven elements be selected in order from a set with four elements when repetition is allowed?

11. How many ways are there to assign three jobs to twenty employees if each employee can be given more than one job?

12. How many solutions are there to the equationx1+x2+x3+x4+x5 =21where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that

a) x1 1?

b) x1 3 for i= 1,2,3,4,5?

c) 0 x1 3, 1 x2 < 4, and x3 15?

d) 0 x1 3, 1 x2 < 4, and x3 15? 13. How many ways are there to distribute thirteen indistinguishable balls into eight distinguishable bins?

14. How many different strings can be made from the letters in MISSISSIPPI. using all the letters?

15. What is the probability that a five-card poker hand contains the nine of diamonds, the eight of clubs and the king of spades?

16. What is the probability that a fair die never comes up an even number when it is rolled four times?

17. Find the probability of selecting none of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding 5O.

18. What is the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing if 48 people enter a contest and


1. Which of these are propositions? What is the truth value of those that are propositions?a) Answer this question. b) What time is it?c) Miami is the capital of Florida. d) 6+x=9e) The moon is made of green cheese.f) 2n130

#2. Let p, q, and r be the propositionsp: You have the flu.q: You miss the final examination.r: You pass the course.Express each of these propositions as an English sentence.

a) qp You miss the final examination if you do not have the flu.

If you do not miss the final examination, then you do not have the flu.

If you miss the final examination, then you have the flu.

If you miss the final examination, then you do not have the flu.

If you do not miss the final examination, then you have the flu.

b) qp You do not miss the final examination if and only if you do not have the flu.

You do not miss the final examination if and only if you have the flu.

You do not miss the final examination if you have the flu.

You miss the final examination if you do not have the flu.

You miss the final examination if and only if you do not have the flu.

c) pqr

You have the flu, or miss the final exam, or pass the course.

You have the flu, and miss the final exam, and pass the course.

You have the flu, and miss the final exam, or pass the course.

You have the flu, or miss the final exam, and pass the course.

You do not have the flu, or miss the final exam, or pass the course.

d) (pr)(qr) If you do not pass the course, then you have the flu and you missed the final exam.

It is the case that if you do not pass the course, then you have the flu or missed the final exam.

It is the case that if you have the flu and miss the final exam, then you do not pass the course.

It is either the case that if you have the flu then you do not pass the course or the case that if you miss the final exam then you do not pass the course.

It is the case that if you have the flu then you do not pass the course and the case that if you miss the final exam then you do not pass the course.

e) (pq)(qr)

Either you have the flu or miss the final exam, or you do not miss the final exam and pass the course.

Either you have the flu and miss the final exam, or you do not miss the final exam and pass the course.

You have both the flu and miss the final exam, and do not miss the final exam and pass the course.

You have either the flu or miss the final exam, or you do not miss the final exam or pass the course.

You have the flu or miss the final exam, and you do not miss the final exam or pass the course.

#3. State the converse, contrapositive, and inverse of each of these conditional statements.

a) If it snows today, I will ski tomorrow.

b) I come to class whenever there is going to be a quiz.

c) A positive integer is a prime only if it has no divisors other than 1 and itself.

#4. Completethe truth table for each of these compound propositions.a) p(qr)b) p(qr)c) (pq)(pr)d) (pq)(qr)e) (pq)(qr)

#5. Are these system specifications consistent?

"Whenever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can save new files. If users cannot save new files, then the system software is not being upgraded."

Let the following statements be represented symbolically as shown:u: "The software system is being upgraded."a: "Users can access the file system."s: "Users can save new files."

Write each system specification symbolically.

"Whenever the system software is being upgraded, users cannot access the file system."

ua

ua

ua

au

ua

"If users can access the file system, then they can save new files."

as

as

sa

as

as

"If users cannot save new files, then the system software is not being upgraded."

us

su

su

su

su

Is the system consistent?

No, this system is not consistent.

Yes, for example makingv false,a false, and s true makes it consistent.

Yes, the conditional statements are always true.

#6. An explorer is captured by a group of cannibals. There are two types of cannibals those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless she can determine whether a particular cannibal always lies or always tells the truth. She is allowed to ask the cannibal exactly one question.

a) Explain why the question Are you a liar? does not work.

Both types of cannibals will answer with "no".

Both types of cannibals will answer with "yes".

The incorrect conclusion that the cannibal is one who always tells the truth will be made if the answer is "no".

The incorrect conclusion that the cannibal is one who always tells the truth will be made if the answer is "yes".

b) Which of the following questions does work in determining whether the cannibal she is speaking to is a truth teller or a liar?

Select all the questions that work.

If I were to ask you if you always told the truth, would you say that you did?

If I say that you are a truth teller, would I be correct?

Do you always tell the truth?

Is the color of the sky blue?

If I say that you are a liar, would I be correct?

#7. Use De Morgan's laws to find the negation ofthe following statement.James is young and strong.

James is young or he is not strong.

James is not young and he is not strong.

James is young or he is strong.


James is not young or he is not strong.

#8. Show that each of these conditional statements is a tautology by completing the truth tables.

a) (pq)q

pqpq(pq)qTT

TF

FT

FF

b) p(pq)

pqpqp(pq)TT

TF

FT

FF

c) p(pq)

pqppqp(pq)TT

TF

FT

FF

d) (pq)(pq)

pqpqpq(pq)(pq)TT

TF

FT

FF

e) (pq)p

pqpq(pq)(pq)pTT

TF

FT

FF

f) (pq)q

pqpq(pq)q(pq)qTT

TF

FT

FF

We conclude that each of these conditional statements is a tautology because____________

#9. Use set builder notation to give a description of each of these sets.

a) {0,4,8,12,16}

{4n|nZ}

{4n|n16}

{n|n16}

{4n|n=1,2,3,4}

{4n|n=0,1,2,3,4}

b) {2, 1, 0, 1, 2}

{x|2x2}, where the domain is the set of integers.

{x|2x2}

{x|2k.

Determine whether each of these functions is O(x^2).

8. To establish a big-Orelationship, find witnesses C and k such that |f(x)|C|g(x)| whenever x>k.


#1. Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.

#2. The intersection graph of a collection of sets A1, A2, . . . , An is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Select the intersection graph of these collections of sets.

#3. Find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. Identify all isolated and pendant vertices.

v=e=

Enter the degree of each vertex as a list separated by commas, starting from vertex a and proceeding in alphabetical order.

Enter the isolated vertices as a list separated by commas, starting from vertex a and proceeding in alphabetical order. Enter NA if there is no isolated vertex.

Enter the pendant vertices as a list separated by commas, starting from vertex a and proceeding in alphabetical order. Enter NA if there is no pendant vertex.

#4. Determine the number of vertices and edges and find the in-degree and out-degree of each vertex for the given directed multigraph.

v=e=deg(a)=deg+(a)=deg(b)=deg+(b)=deg(c)=deg+(c)=deg(d)=deg+(d)=

#5. Use an adjacency list to represent the given graph.

Enter the vertices in alphabetical order, separated by commas.VertexTerminal Verticesabcd

#6. Use an adjacency matrix to represent the given graph.Assume the vertices are listed in alphabetical order.

#7. Which of the following graphs has the given adjacency matrix?

MTH 221 Week 5 Final Exam1. A particular brand of shirt comes in8 colors, has a maleversion and a female version, and comes in5 sizes for each sex. How many different types of this shirt are made?

2. A club has 22 members.

a) How many ways are there to choose four members of the club to serve on an executive committee?b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?

3. In how many different ways can ten elements be selected in order from a set withfour elements when repetition is allowed?

4. What is the probability that a fair die never comes up anodd number when it is rolledeight times?

5. Let p, q, and rbe the propositions

p:

You have the flu.

q:

You miss the final examination.

r:

You pass the course.

Express each of these propositions as an English sentence.

a)qr

You miss the final examinationifyou pass the course.

If you do not miss the final examination, then you pass the course.

If you miss the final examination, then you pass the course.

If you do not miss the final examination, then you do not pass the course.

If you miss the final examination, then you do not pass the course.

b)qp

You miss the final examination if and only if you do not have the flu.

You do not miss the final examinationif you have the flu.

You do not miss the final examinationif and only if you do not have the flu.

You do not miss the final examinationif and only if you have the flu.

You miss the final examinationif you do not have the flu.

c)pqr

You have the flu, or miss the final exam, and pass the course.

You do not have the flu, or miss the final exam, or pass the course.

You have the flu, and miss the final exam, and pass the course.

You have the flu, and miss the final exam, or pass the course.

You have the flu, or miss the final exam, or pass the course.

d)(pr)(qr)

It is the case that if you have the fluand miss the final exam, then you do not pass the course.

It is the case that if you have the flu then you do not pass the course and the case that if you miss the final exam then you do not pass the course.

It is the case that if you do not pass the course, then you have the flu or missed the final exam.

It is either the case that if you have the flu then you do not pass the course or the case that if you miss the final exam then you do not pass the course.

If you do not pass the course, then you have the flu and you missed the final exam.

e) (pq)(qr)

You have either the flu or miss the final exam, or you do not miss the final exam or pass the course.

Either you have the flu and miss the final exam, or you do not miss the final exam and pass the course.

You have both the flu and miss the final exam, and do not miss the final exam and pass the course.

Either you have the flu or miss the final exam, or you do not miss the final exam and pass the course.

You have the flu or miss the final exam, andyou do not missthe final exam or pass the course.

6. Completethe truth table for each of these compound propositions.

a) p(qr)pqrqqrp(qr)TTT

TTF

TFT

TFF

FTT

FTF

FFT

FFF

b) p(rq)

pqrprqp(rq)TTT

TTF

TFT

TFF

FTT

FTF

FFT

FFF

c) (pq)(pr)

pqrppqpr(pq)(pr)TTT

TTF

TFT

TFF

FTT

FTF

FFT

FFF

d) (pq)(qr)

pqrqpqqr(pq)(qr)TTT

TTF

TFT

TFF

FTT

FTF

FFT

FFF

e) (pq)(qr)

pqrpqpqqr(pq)(qr)TTT

TTF

TFT

TFF

FTT

FTF

FFT

FFF

7. Use De Morgan's laws to find the negation ofthe following statement.

James is young and strong.

James is young or he is strong.



James is young or he is not strong.

James is not young or he is not strong.

8. Show that each of these conditional statements is a tautology by completing the truth tables.

a) (pq)ppqpq(pq)pTT

TF

FT

FF

b) q(pq)

pqpqq(pq)TT

TF

FT

FF

c) p(pq)

pqppqp(pq)TT

TF

FT

FF

d) (pq)(pq)

pqpqpq(pq)(pq)TT

TF

FT

FF

e) (pq)p

pqpq(pq)(pq)pTT

TF

FT

FF

f) (pq)q

pqpq(pq)q(pq)qTT

TF

FT

FF

We conclude that each of these conditional statements is a tautology because the entries in the last columncontain_________.

9. Suppose that A={6,7,8}, B={3,6,7}, C={3,7}, and D={6,7}. Determine which of these sets are subsets of which other of these sets.

A

A

B

C

D

B

A

B

C

D

C

A

B

C

D

D

A

B

C

D

10. Use a Venn diagram to illustrate the relationship ABand BC.

11.Suppose that Ais the set of sophomores at your school and Bis the set of students in discrete mathematics at your school. Express each of these sets in terms of Aand B.

a) the set of sophomores at your school who are not taking discrete mathematics

AB

ABAB

ABAB

b) the set of students at your school who either are sophomores or are taking discrete mathematics

AB

AB

ABAB

AB

12. Let A={a,b,c,d,e}and B={a,b,c,d,e,f,g,h}. Find

a) AB

{}

{f,g,h}

{a, b, c, d, e}

{a, b, c, d, e, f, g, h}

{a,b,c,d}

b) AB

{}

{a,b,c,d,e,f,g,h}

{a,b,c,d,e}

{e,f,g,h}

{f, g, h}

13. Select the correct Venn diagram for each of these combinations of the sets A, B, and C.

a) B(AC)

b) ABC

c) (BA)(CB)(AC)

MTH 221 Week 5 Case StudyIndividual Case Study Applications PaperChoose one of the following Case Studies:

Food Webs

Coding Theory

Network Flows

Write a 750- to 1,250-word paper in which you complete one of the following options:

Option 1: Food Webs Case Study

Explain the theory in your own words based on the case study and suggested readings.

Include the following in your explanation:

Competition

Food Webs

Boxicity

Trophic Status

Give an example of how this could be applied in other real-world applications.

Format your paper according to APA guidelines. All work must be properly cited and referenced.

MTH/221

Discrete Math for Information Technology

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