Please help answer distrecte statistics homework – questions located in file attached

Please help answer discrete statistics homework – questions located in file attached:

 

Attached.

 

1

Exam II (The exam is due Wednesday, October 30th by 8am) Discrete Structures I

Directions: Exam 1I is due Wednesday, October 30th, by 8am. Answer all questions
carefully. Do all steps with detailed explanations. I prefer that you send your solutions
electronically. They need not be typed. Follow this procedure:

1) On the course home page, click on Exam Dropboxes folder
2) Click on Exam II dropbox
3) Scroll down and next to “Add Attachments” click the “Browse My Computer” button to attach your
exam. Note, make sure you to attach your exam in Microsoft Word format or as a PDF file if scanned.)
4) Click Submit to send your exam to me.
Do not use the web board to submit exams. If you wish to FAX or mail your answers to me they

again, need not be typed but I must receive them by the due date and time. Make sure the
course number and name (92.321, Discrete Structures I) as well as your name and my name
appear on the top of each answer sheet. I must receive all exams by the due date. Use the
cover sheet for the FAX ((978) 934-4064). To mail exams to me send them to:

Prof. Alan W. Doerr
Mathematical Sciences Department
UMASS Lowell
I University Ave.
Lowell, Mass 01854

Good luck on your exam.

1. (20 pts.) Note the contrapositive of the definition of one-to-one function given on
page 141 of the text is: If a ≠ b then f(a) ≠ f(b). As we know, the contrapositive is
equivalent to (another way of saying) the definition of one-to-one.
(a) Consider the following function f: R → R defined by f(x) = x2 + -5x + 6

.

Use the contrapositive of the definition of one-to-one function to determine (no proof
necessary) whether f is a one-to-one function. Explain

(b) Compute f  f.

(c) Let g be the function g: R → R defined by g(x) = x3 + 5. Find g -1. Use the
definition of g -1 to explain why your solution, g -1 is really the inverse of g.

2. (5 points) Text page 169 number 34, part a.

3.(15 points) Let A =
1

1 1

2 1 1
1 1 2

− − 
 − 
 − − − 

, B =
1 1 1
2 2 1

2 1 3

− 
 − 
  

2

and C =
2 2
1 0

1 1
− 
 − 
  

Compute:
(a) AC + BC (It is much faster if you use the distributive law for matrices first.)

(b) 2A – 3A

(c) Perform the given operation for the following zero-one matrices. See the
text, page 182, for the definition of the symbol .

1 1 1 1 1 1
1 0 0 0 1 1
0 1 1 0 0 1

   
   
   
      



4. (10 points) Let A =
2 1
2 2
 
 
 

.

(a) Use the formula for finding the inverse of a 2×2 matrix given in exercise 19
page 184 to find A-1 .
(b) Use the definition of the inverse of a matrix given in the text just prior to
exercise 18, page 184, to do #18.

Read your notes carefully before attempting problems 5 and 6. Simply follow the
examples). You must show all work. Here are a couple of online sites if you need
them: Klan Academy Videos (click on Linear Algebra) and google Gaussian
Elimination.

5. (15 points) Solve the following systems of equations using the method outlined in
week 7 (Part I) of the notes. Your procedure should be in matrix form as was done in
Example 3.

x1 + x2 + x3 = 1
2×1 – x2 + x3 = 2
-1×2 + x3 = 1

6. (10 points) Week 7 (Part II, Systems Which Do Not Have a Unique Solution) of the
notes. Page 5 number 2.

8. (10 points.) We know that matrix algebra behaves similar to
(but not exactly the same as) regular algebra. The statements in parts a and b illustrate a
couple of the differences between the two structures.
Let A and B be arbitrary n x n matrices whose entries are real numbers.

(a) Use basic matrix laws only to expand (A + B )2. Explain all steps. Note the
basic matrix laws are given in your notes, week 6 pg.4. Hint: Use the
distributive laws.

3

(b) Is (A – B)(A + B) = A2 – B2 ? Explain as you did in part (a).

9. (15 pts) Read pages 311 to 318 and study examples 1, 2 and 3. Then do

Text, page 329, number 4.

Ext ra Point s

i. Solve f or B:
2 1 3 0

B =
2 2 1 2
   
   
   

. Hint , one way is t o use problem 4 a.

ii. Text page 1 5 3 number 1 2 part s c and d. Explain, but no proof necessary.

iii. This problem can be used for extra credit for either Exam 2 or exam 3.
Everyone should try it. It’s a nice application. It’s is long so it is worth 15 extra points.

Hand it in for exam 2 when exam 2 is due or for exam 3 (when exam 3 is due)
Project: Make up a meaningful network example. In the notes, week 7 part III,

on Network Analysis I have given you several different “flow models”. Your project
does not have to mimic one of these. Try to make it different than those presented in the
notes.
a) Your example must be well defined. Define the problem clearly.

b) Your example should involve solving a system of equations with an “infinite
number of solutions”. Actually, once you place restrictions on the variables (They may
be integers and see part f.) the number of solution may be large but not an infinite
number.

c) Solve your system of equations any way you wish, but describe your solutions
clearly.

d) Explain your results with two or three examples. That is, compute the flow
when the free variable are specific numbers.

e) Interpret your results. Do they make sense? Do negative values make sense?
f) Should there be a maximum capacity for each edge? If so, make up some
reasonable maximum capacity for each edge.