mth 221 entire class tutorials

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MTH/221 Week 1 - Connect Exercises

Combinatorics

1.1 Apply basic enumeration techniques.

1.2 Apply basic permutation and combination techniques.

1.3 Apply introductory probability techniques.

Week 1 DQ#1- What is the difference between combinations and permutations? What are some practical applications of combinations? Permutations?

Week 1 DQ#2 -Find the number of permutations of A,B,C,D,E,F taken three at a time (in otherwords find the number of "3-letter words" using only the given six letters WITHOUT repetition).

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Week 1 DQ#3 -Suppose you are assigning 6 indistinguishable print jobs to 4 indistinguishable printers. In how many ways can the print jobs be distributed to the printers?

Week 1 DQ#4 - Suppose UOPX has 3 different math courses, 4 different business courses, and 2different sociology courses. Tell me the number of ways a student can choose one of EACH kind ofcourse. Then tell me the number of ways a student can choose JUST one of the course.

Week 1 DQ#5 - Consider the problem of how to arrange a group of n people so each person canshake hands with every other person. How might you organize this process? How many times willeach person shake hands with someone else? How many handshakes will occur? How must yourmethod vary according to whether or not n is even or odd?

1. 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 digits

1. do not contain the same digit twice? 2. end with an even digit? 3. have exactly four digits that are 9’s?

3. How many strings of six uppercase English letters are there

1.

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 anexecutive committee?



b) How many ways are there to choose a president. vice president. secretary. andtreasurer 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 school

a) How many ways are there to select a committee of five members of the department if at least onewoman 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 equation

x1+x2+x3+x4+x5 =21

where 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 eightdistinguishable bins?

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



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

ALL DETAILED ANSWERS PROVIDED


Logic & Set Theory; Boolean Algebra;Relations & Functions

2.1 Use truth tables for propositional logic.

2.2 Simplify assertions and compound statements in first-order logic.

2.3 Apply basic set-theoretic concepts.

2.4 Use a Venn Diagram to visualize set relationships.










2.5 Differentiate between relations and functions.

2.6 Apply the basic concepts of Boolean algebra.

Week 2 DQ#1- There is an old joke, commonly attributed to Groucho Marx, which goes somethinglike this: "I don't want to belong to any club that will accept people like me as a member." Does thisstatement fall under the purview of Russell's paradox, or is there an easy semantic way out? Look

up the term fuzzy set theory in a search engine of your choice or the University Library, and see ifthis theory can offer any insights into this statement.

Week 2 DQ#2- How do we distinguish relations from functions?

Week 2 DQ#3- What sort of relation is friendship, using the human or sociological meaning of theword? Is it necessarily reflexive, symmetric, antisymmetric, or transitive? Explain why it is or is not

any of these. What other types of interpersonal relationships share one or more of these properties?Explain.

Week 2 DQ#4 - Write the dual of the following Boolean equation: a+a'b = a+b?

Week 2 DQ#5- Reduce the following Boolean product to zero OR a fundamental product: xyx'z.

WEEK 2 CONNECT EXCERCISES

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) 2n≥130

#2. Let p, q, and r be 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) ¬q→¬p

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

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

b) ¬q↔p

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

c) p q r

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

d) (p→¬r)

(q→¬r)

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

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

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

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

e) (p q) (¬q r)

o Either you have the flu or miss the final exam, or you do not miss the final exam and

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

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

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

o You have the flu or miss the final exam, and you do not miss the final exam or passthe 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. Complete the truth table for each of these compound propositions.

a) p→(¬q

r)b) ¬ p→(q→ r)

c) ( p→q) (¬ p→ r)

d) ( p↔q) (¬q↔ r)

e) (¬ p↔¬q)↔(q↔ r)

#5. Are these system specifications consistent?

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

u∧¬a

u→a

u↔¬a

¬a→u

u→¬a

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

a∨s

a∧s

s→a

a↔s

a→s



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

¬u→¬s

¬s∧¬u

¬s↔¬u

¬s→¬u ¬s→u

Is the system consistent?

No, this system is not consistent.

Yes, for example making v 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 whoalways tell the truth and those who always lie. The cannibals will barbecue the explorer unless shecan determine whether a particular cannibal always lies or always tells the truth. She is allowed toask 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 ifthe answer is "no".

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

b) Which of the following questions does work in determining whether the cannibal she isspeaking 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 of the 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) ( p q)→q

q p q ( p q)→q

T T

T F

F T

F F

b) p→( p q)

q p q p→( p q)

T T

T F

F T

F F

c) ¬ p→( p→q)

q ¬ p p→q ¬ p→( p→q)

T T

T F

F T

F F



d) ( p q)→( p→q)

q p

q p→q ( p q)→( p→q)

T T

T F

F T

F F

e) ¬( p→q)→ p

q p→q ¬( p→q) ¬( p→q)→ p

T T

T F

F T

F F

f) ¬( p→q)→¬q

q p→q ¬( p→q) ¬q ¬( p→q)→¬q

T T

T F

F T

F F

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|n∈Z}

{4n|n≤16}

{n|n≤16}

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

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



b) {−2, −1, 0, 1, 2}

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

{ x|−2≤ x≤2}

{ x|−2< x<2}, where the domain is the set of real integers.

{ x|−2≤ x≤2}, where the domain is the set of real numbers.

{ x|−2≤ x≤2}, where the domain is the set of natural numbers.

c) { a, b, c, r, s}

{ x| x is a letter of the word scrabble other than l or e}

{ x| x is a letter}

{ x| x is a letter of the word scrabble}

{ x| x is a letter of the word scrabble other than l}

{ x| x is a letter of the word scrabble other than e}

#10. Determine whether each of these pairs of sets are equal.

a) {1, 1, 3, 3, 5, 5, 5, 6},{5, 3, 1}b) {{3}},{3,{3}}c) ∅,{∅}

#11. Suppose that A={2,5,6}, B={2,6}, C ={5,6}, and D={5,6,7}. 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

#12. Use a Venn diagram to illustrate the relationship A⊆ B and B⊆C .





Algorithmic Concepts

• 3.1 Apply the basic concepts of algorithmic analysis. • 3.2 Apply the introductory principles of mathematical induction. • 3.3 Solve problems of iteration and recursion.

Week 3 DQ#1 - Describe a situation in your professional or personal life when recursion, or at leastthe principle of recursion, played a role in accomplishing a task, such as a large chore that could bedecomposed into smaller chunks that were easier to handle separately, but still had the semblance of

the overall task. Did you track the completion of this task in any way to ensure that no pieces wereleft undone, much like an algorithm keeps placeholders to trace a way back from a recursivetrajectory? If so, how did you do it? If not, why did you not?

Week 3 DQ#2 - Given this recursive algorithm for computing a factorial...

procedure factorial(n: nonnegative integer)

if n = 0 then return 1

else return n *factorial(n − 1)

{output is n!} Show all the steps used to find 5!

Week 3 DQ#3 - Describe an induction process. How does induction process differ from a processof simple repetition?










Week 3 DQ#4 - List all the steps used to search for 9 in the sequence 1,3, 4, 5, 6, 8, 9, 11 using a binary search.

WEEK 3 CONNECT EXERCISES

1. List all the steps used by Algorithm 1 to find the maximum of the list

3,9,14,7,11,4,18,3,11,2

2. Determine which characteristics of an algorithm described in the text (after Algorithm 1) thefollowing procedures have and which they lack. Select characteristics that the procedures have andleave characteristics unselected that the procedures lack.

a. procedure double(n: positive integer)

while n > 0

n = 3n

b. procedure divide(n: positive integer)

while n ≥ 0

m = 1 / n n = n − 1

c. procedure sum(n: positive integer)

sum = 0while I < 3sum =sum + i

d. procedure choose(a,b: integers)

x = either a or b

3. Which of the following algorithms can be used to find the sum of all the integers in a list?

4. Which statement best describes an algorithm that takes as input a list of n integers and producesas output the smallest difference obtained by subtracting an integer in the list from the onefollowing it?






Set the answer to be ∞. For i going from 1 through n−1, compute the

value of the (n+1)st element in the list minus the nth element in thelist. If this is smaller than the answer, reset the answer to be thisvalue.

Set the answer to be 0. For i going from 1 through n, compute thevalue of the (i+1)st element in the list minus the ith element in the list.If this is smaller than the answer, reset the answer to be this value.

Set the answer to be −∞. For i going from 1 through n−1, compute thevalue of the (i+1)st element in the list minus the ith element in the list.If this is smaller than the answer, reset the answer to be this value.

Set the answer to be0. For i going from 1 through n−1, compute the

value of the (i+1)st element in the list minus the ith element in the list.If this is smaller than the answer, reset the answer to be this value.

Set the answer to be ∞. For i going from 1 through n−1, compute the

value of the (i+1)st element in the list minus the ith element in the list.If this is smaller than the answer, reset the answer to be this value.

5. Consider an algorithm that uses only assignment statements that replaces the hextuple (u, v, w, x,

y, z) with (v, w, x, y, z, u). What is the minimum number of assignment statements needed?

6. Which of the following is an algorithm that locates the last occurrence of the smallest element ina finite list of integers? The integers in the list are not necessarily distinct.

7. To establish a big-O

relationship, find witnessesC

andk

such that | f

( x

)|≤C

|g( x

)| whenever x

>k .

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 .

PLUS QUESTIONS 9 - 18 WITH DETAILED

EXPLANATIONS ON HOW ANSWERS ARE OBTAINED

FOR EACH



MTH/221 Week 4 -Connect Exercises

Graph Theory and Trees

• 4.1 Apply properties of general graphs. • 4.2 Apply properties of trees.

WEEK 4 DQ#1 - Hamiltonian and Euler Graphs

Note a Hamiltonian circuit visits each vertex only once but may repeat edges. A Eulerian graphtraverses every edge once, but may repeat vertice.

Looking at the figure below tell me if they are Hamiltonian and/or Eulerian.

*-------*--------*| / | \ || / | \ ||/ | \ |*-------*--------*

WEEK 4 DQ#2 - Path Analysis Class

Here is a Question for you:For the diagram below find all the "simple paths" from A to F.

A------------B------------C| | / || | / || | / |

D------------E------------F









WEEK 4 DQ#3 - Random Graphs

Random graphs are a fascinating subject of applied and theoretical research. These can be generatedwith a fixed vertex set V and edges added to the edge set E based on some probability model, suchas a coin flip. Speculate on how many connected components a random graph might have if thelikelihood of an edge (v1,v2) being in the set E is 50%. Do you think the number of components

would depend on the size of the vertex set V? Explain why or why not.

WEEK 4 DQ#4 - Trees and Language Processing

Trees occur in various venues in computer science: decision trees in algorithms, search trees, and soon. In linguistics, one encounters trees as well, typically as parse trees, which are essentially

sentence diagrams, such as those you might have had to do in primary school, breaking a natural-language sentence into its components--clauses, subclauses, nouns, verbs, adverbs, adjectives, prepositions, and so on. What might be the significance of the depth and breadth of a parse treerelative to the sentence it represents? If you need to, look up parse tree and natural language processing on the Internet to see some examples.

WEEK 4 CONNECT EXERCISES -QUESTIONS AND ANSWERS TO #1 to #14

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








#2. The intersection graph of a collection of sets A1, A2, . . . , An is the graph that has a vertex foreach of these sets and has an edge connecting the vertices representing two sets if these sets have anonempty 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 givenundirected 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 proceedingin alphabetical order.

Enter the isolated vertices as a list separated by commas, starting from vertex a and proceeding inalphabetical 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 inalphabetical 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 Vertices

a bcd

#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?

INCLUDES REMAINING QUESTIONS AND

ANSWERS FROM #8 to #14
















MTH/221 - Week 5 - Final Exam & Discussion

Questions

Applications of Discrete Mathematics

Option 2: Coding Theory Case Study

Explain the theory in your own words based on the case study

and suggested readings.

Include the following in your explanation:

• Error Detecting Codes• Error Correcting Codes

• Hamming Distance

• Perfect Codes

• Generator Matrices

• Parity Check Matrices

• Hamming Codes

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.

DISCRETE MATH - FINAL EXAM -

CONNECT EXERCISES








1. A particular brand of shirt comes in 8 colors, has a male version and a female

version, and comes in 5 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 executivecommittee?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

with four elements when repetition is allowed?

4. What is the probability that a fair die never comes up an odd number when it is

rolled eight times?

5. Let p, q, and r be the propositions

: 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) ¬q→r

You miss the final examination if you 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) ¬q↔ p

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

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

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



c) p∨q∨r

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) ( p→¬r )∨(q→¬r )

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

It is the case that if you have the flu then you do not pass the course and the case that if youmiss 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 finalexam.

It is either the case that if you have the flu then you do not pass the course or the case that ifyou 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) ( p∧q)∨(¬q∧r )

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

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

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

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

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

6. Complete the truth table for each of these compound propositions.

a) p→(¬q

r)

p q r ¬q ¬q∧r p→(¬q

r)

T T T



T T F

T F T

T F F

F T T

F T F

F F T

F F F

b) ¬ p→( r→q)

q r ¬ p r →q ¬ p→( r→q)

T T T

T T F

T F T

T F F

F T T F T F

F F T

F F F

c) ( p→q) (¬ p→ r)

q r ¬ p →q ¬ p→r ( p→q) (¬ p→ r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

d) ( p↔q) (¬q↔ r)

q r ¬q ↔q ¬q↔r ( p↔q) (¬q↔ r)

T T T

T T F

T F T

T F F

F T T

F T F



F F T

F F F

e) (¬ p↔¬q)↔(q↔ r)

q r ¬ p ¬q ¬ p↔¬q q↔r (¬ p↔¬q)↔(q↔ r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

7. Use De Morgan's laws to find the negation of the 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) ( p q)→ p

p q p q ( p q)→ p

T T

T F

F T

F F

b) q→( p q)

q p

q q→( p

q)



T T

T F

F T

F F

c) ¬ p→( p→q)

q ¬ p p→q ¬ p→( p→q)

T T

T F

F T

F F

d) ( p q)→( p→q)

q p q p→q ( p q)→( p→q)

T T

T F

F T

F F

e) ¬( p→q)→ p

q →q ¬( p→q) ¬( p→q)→ p

T T

T F

F T

F F

f) ¬( p→q)→¬q

q →q ¬( p→q) ¬q ¬( p→q)→¬q

T T

T F

F T

F F



We conclude that each of these conditional statements is a tautology because the entries in the lastcolumn contain _________.

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

BC

D

D⊆

A

B

C

D

10. Use a Venn diagram to illustrate the relationship A⊆ B and B⊆C .




11. Suppose that A is the set of sophomores at your school and B is the set of

students in discrete mathematics at your school. Express each of these sets in terms

of A and B.

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

A∩ B —

A∪ B

A∪ B —

A — ∩ B

A∩ B

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

A — ∪ B —

A∪ B — A∪ B

A — ∩ B —

A∩ B

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

a) A∩ B







{}

{ f ,g,h}

{a, b, c, d, e}

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

{a,b,c,d }

b) A− B

{}

{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∪( A∩C )






b) A∩ B — ∩C








c) ( B− A)∪(C − B)∪( A−C )








INCLUDES REMAINING QUESTIONS AND

ANSWERS FROM #14 to #40

Week 5 DQ#1 - After performing some research or based on your reading in the course, share withthe class the most practical use of discrete mathematics (in your opinion). Please cite your source.

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