CSCE 235 HW 4

CSCE 235

Homework Assignment 4 (with Programming)

Assigned: February 6, 2009

Due: 12:30 p.m. February 16, 2009

Set Basics

1. (10 points) List all members of the following sets.

(a)

(b)

(d) where is the set of all positive numbers

(e) where is the set of all positive numbers

2. (10 points) Consider the following six subsets of Z. (Definition: Z = the set of all integers = ( …, -3, -2, -1, 0, 1, 2, 3, …}.)

A = { 2m + 1 | m Î Z }

B = { 2n + 3 | n Î Z }

C = { 2p -3 | p Î Z }

D = { 3r + 1 | r Î Z }

E = { 3s + 2 | s Î Z }

F = { 3t - 4 | t Î Z }

Which of the following statements are true and which are false? Justify your answers.

(a) A = B

(b) A = C

(d) D = E

(e) D = F

(f) E = F

Set Operations

3. Application of Knowledge (21 points) Consider an alphabet Σ = { a, b }. It has only two letters. A word (or w) is any finite string of letters that you can form out of the alphabet. For example, abba is a word formed using only the letters from the alphabet Σ. The collection of all words is called a language, and is denoted as . For example, = { ε, a, aa, ab, ba, bb, aaa, aab, abb, bbb, bba, baa, …}. And is an infinite set! The symbol ε denotes the empty word or null word.

Now, suppose Σ = { a, b }; A = {a, b, aa, bb, aaa, bbb }; B = { : length(w) ≥ 2}; and C = { : length(w) ≤ 2}. (Hint: Don’t forget the empty word ε, which has a length of zero!) List all members of the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g) power set of Σ

4. (15 points) True or False (if false, show an example; if true, explain):

(a) for all A, B, and C.

(b) If A = B, then is .

(d) whenever .

(e) for all A and B.

5. (9 points) Prove the following formally, without using Venn diagrams:

(a) (5 points) for any sets of A, B, and C. (Hint: Use the fact .)

(b) (4 points) If and , then .

Binary Relations

6. (20 points) Determine whether or not each of the binary relations defined on the given sets A are reflexive, symmetric, or transitive. If a relation has a certain property, prove this is so; otherwise, provide a counterexample to show that it does not. Determine also whether a relation is equivalence.

(a) A = { 3, 4 }; = { (3, 3), (4, 4)}

(b) A is the set of all humans; if and only if where means “a is the parent of b.”

(d) A is the set of cross products of integers, Z×Z; if and only if .

(e) A is the set of all humans; if and only if where means “a and b are both fans of the same musician.” Assume that a human may be a fan of more than one musician.

7. (24 points) Given the following relations:

List 5 members for each of the following sets and justify; if not possible, explain.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Equivalence Relations and Partitions

8. (6 points) Let A be the set of all students in the CSCE235 class. For , define when a is taller than or as tall as b. Is an equivalence relation? Justify your answer.

9. (15 points) A partition of a set A is a collection of disjoint nonempty subsets of A whose union is A. These disjoint sets are called cells (or blocks). Suppose that A has the following cells: m, n, o, and p.

(a) What is ?

(b) What is ?

(d) What is ?

(e) What is ?

Programming

10. (50 points) Toolbox. Implement a relation-based toolbox to determine whether a binary relation is reflexive, symmetric, or transitive. Your program should be able to read in a text file similar to the following. Each line represents the members of one binary relation. So, in the above example, the binary relation R1 has 5 members, i.e., { (a,a), (b,c), (c,b), (b,b), (c,c) }. Thus, for R1, it is reflexive, symmetric, and transitive.

5 a a b c c b b b c c

6 a b b c c d d c b d d d

8 j m m j g b a m m a b g g l l g

END

Your output should look like this:

reflexive symmetric transitive

n-reflexive n-symmetric n-transitive

n-reflexive symmetric n-transitive

END

Your program will be tested with an input file with the specified format but with different data. (On our course website, there is a C and a Java program on how to read data from a file. If you choose to use any of these programs, please clearly acknowledge the use in your documentation.) Write your program with good documentation and style. Hand in your program before 12:30 p.m., February 16 to our class Handin account. If your program is complicated and involves multiple files, please also include a README file on how to run it. I expect that a single program file will be sufficient for this problem.

CSE Web Handin

Registration
You have to register with Handin before you use it for the first time. To register, choose the "Register" link from the menu on the left. Input your CSE username along with a password (NOT your CSE password) on the registration page, and click the "Register" button at the bottom of the page to finish the process. You will then receive mail on CSE to confirm your registration.

Handing In
To hand in an assignment, go to CSE235 class's handin web page (http://cse.unl.edu/~cse235/handin) and fill out the form on that page. Use your CSE username and your Handin password. Be sure to choose the correct assignment to hand the files in under, and choose the files you wish to hand in with the file choosers at the bottom. If you need to hand in more files than there are selection boxes, hand them in two (or more) separate batches. For example, if you need to hand in the files file1.c, file2.c, file3.c, file4.c, file5.c, and file6.c, but handin only lets you hand in five files at a time, hand in file1.c through file5.c as one group and file6.c by itself. After each handin you will be shown a list of all of the files you have handed in up to that point.

Checking a previous handin
If ever you want to check the status of a previous handin ("Did I really hand in those files?"), you can go to your class's handin page and choose "Check Status" from the handin choices. After making that choice, filling in your username and password, and choosing an assignment to check up on, click the Submit button at the bottom of the page. You will then be presented with a list of all of the files you have handed in for that particular assignment, along with their sizes and the time/day you handed them in.

Changing your password
Changing your password on occasion can be a good idea. To change your password, choose the "Password" link from the menu on the left. Fill out the "Change user password" form with your username, current password, and the new password you would like. The change will take effect immediately after you submit it, and you will also be sent a piece of email on CSE confirming the change.

Forgotten password
If you forget your password, Handin can send it to you at your CSE email address. To receive your password, choose the "Password" link from the menu on the left and fill out the "Send user password" form with your username. After you submit the form you will receive mail on CSE with your Handin password.