A subset of a set is, roughly, a set inside the set. Precisely, a set A is a subset of set B, denoted A $\subseteq$ B, if and only if every element of A is also an element of B. Consider the set N = {2, 4, 5, 6, 8, 11}.

a. Is {5, 2} a subset of N? Why or why not?

b. Is {4, 6, 8, 10} $\subseteq$ N? Why or why not?

c. Write two other subsets of N.

d. Is {2, 4, 5, 6, 8, 11} a subset of N? Justify your answer by using the definition of a subset.

e. In a search for used Chevrolet trucks on the Internet, think about three possible search sets:

• T = {x | x is a Web page containing the word truck}
• U = {x | x is a Web page containing the phrase "used truck"}
• C = {x | x is a Web page containing the word Chevrolet}

Which of these sets is a subset of one of the other sets? Draw a Venn diagram that shows the relationships among these three sets.

Now explore some other modular arithmetic systems.

a. Consider a mod 33 system. Perform the same computations as in the earlier ptoblem, except this time uses mod 33.

b. Consider a mod 15 system.

i. Why does it make sense to say that 23 and 38 are "equivalent mod 15"?

ii. What is 6$^4$ mod 15?

You can build a device like that shown for quickly encrypting (or decrypting) messages using a Caesar cipher. The outer wheel represents the plaintext and the inner wheel represents the ciphertext. The inner wheel can be rotated to different positions.

a. Describe the particular cipher shown here in words and in an equation of form c = p + n mod 26.

b. How many different Caesar ciphers are possible? Explain your reasoning.

c. Experiment with the Caesar Cipher feature of the TCMS-Tools "Codes" custom app.

i. Use the tool to create the cipher in Part a.

ii. Use the tool to illustrate your answer to Part b.

iii. Compare the Caesar Cipher feature to the ROT13 Cipher feature of the "Codes" custom app. Explain how they are similar and how they are different.

d. Experiment with the Affine Cipher feature of the TCMS-Tools "Codes" custom app.

i. Explain in words how the Affine Cipher encrypts.

ii. What settings in the Affine Cipher will create the ROT13 cipher?

iii. Explain why the Affine cipher is not a Caesar cipher. How is this shown in the slope of the respective linear models?

Examine the conjectures below-relating complements of sets.

a. Draw Venn diagrams that provide a "proof without words" for whether each conjecture is true or false.

• Conjecture I: $\quad$ (A $\cap$ B)$^{\prime}$ = A$^{\prime} \cap$ B$^{\prime}$

• Conjecture II: $\quad$ (A $\cap$ B)$^{\prime}$ = A$^{\prime} \cup$ B$^{\prime}$

b. Write and test a conjecture, similar to those above, that begins (A $\cup$ B)$^{\prime}$ = _____________. Compare your conjecture with those of your classmates. Resolve any differences.

c. For each true conjecture in Parts a and b, write the conjecture in words.

Now use the RSA cryptosystem to send and receive secret messages. Working with your teacher, form teams with your classmates. Choose a name for your team. Your team should choose one of the public encryption keys listed in the public-key directory shown below. Write your team name next to your public key on a copy of the Public-Key Directory. Your teacher will hand each group their corresponding private key. Thus, each team has its own pair of keys-a public key and the associated private key.

Public-Key Directory

$$\begin{array}{|c|c|c|} \hline \text { Team Name } & \boldsymbol{n} & e \\ \hline & 85 & 13 \\ \hline & 55 & 23 \\ \hline & 161 & 19 \\ \hline & 95 & 31 \\ \hline & 91 & 29 \\ \hline & 65 & 29 \\ \hline & 35 & 13 \\ \hline \end{array}$$

a. Your team should use the RSA cryptosystem to encrypt and send at least one single-word secret message to another team. As per the cryptosystem rules, use the receiving team's public key to encrypt the message.

b. Decrypt any messages you receive. As per the cryptosystem rules, use your own private key to decrypt the message you receive.

c. Based on this public-key cryptosystem, discuss and answer the following questions.

i. Can any team send any other team a secret encrypted message? What information do you need to encrypt a secret message so that only the target team can read it?

ii. Once you encrypt a secret message to a target team, can any other team decrypt it and read the message? Explain.

iii. Suppose you receive an encrypted message from another team. What information did that team need to encrypt the message? What information do you need to decrypt it? Can any other team decrypt the message?

The ciphers in the earlier problems are called substitution ciphers because of the substitution technique used. A cipher is an algorithm used for encryption and decryption. Generally, there are two related algorithms, one for encryption and one for decryption. A substitution cipher is an algorithm that encrypts a message by replacing each character in the plaintext with another character. The receiver reverses the substitution to decrypt the message. A key typically consists of the information needed to carry out the encryption (or decryption) process. In a shift substitution cipher, a key is typically a number. For example, the key in a Caesar cipher is the number of places down the alphabet that you must shift.

a. What is the key for the Caesar cipher in the earlier problem?

b. What is the key for the ROT13 cipher?

c. Substitution ciphers like ROT13 and Caesar ciphers are not very secure. That is, it is not very difficult to decrypt the ciphertext even if you do not know the key. Describe some general strategies you think could be used to break substitution ciphers like those above.

d. Create your own substitution cipher. Encrypt a brief message. Then exchange encrypted messages with a classmate and try to decrypt each other's messages. Discuss and resolve any difficulties in the decryption process.

Examine each conjecture below involving operations on sets A and B.

a. Explain or illustrate why the conjecture is true or false. For example, you might draw Venn diagrams to give a "proof without words" for whether it is true or false.

• Conjecture I: $\quad$ A - B = A $\cap$ B$^{\prime}$
• Conjecture II: $\quad$ A - B$^{\prime}$ = A $\cap$ B

b. For each conjecture that is true, describe the statement in words.

The tenth digit that you found in the earlier problem is the check digit. It is used to detect and correct errors that may occur when the ZIP code is read, recorded, or transmitted by a computer. In this problem, you will discover how the check digit works.

a. Add up all nine digits in the ZIP code on the reply card, and also add the tenth check digit. Then reduce mod 10.

b. Repeat Part a for the following three additional ZIP codes, with indicated check digits.

i. 47402-9961, with check digit 8

ii. 80323-4506, with check digit 9

iii. 02174-4131, with check digit 7

c. By examining the results in Parts a and b, state a rule for how the check digit works. Compare your rule with that of others and resolve any differences.

Complete the following tasks related to set theory and information processing.

a. Suppose S = {2, 3, 9, 4, -2} and T = {3, 6, 9, 12}.

i. Find S $\cup$ T, S $\cap$ T, and S - T.

ii. Draw Venn diagrams illustrating the operations in part i.

iii. Let V = {x | x = 6 n, where n is a positive integer}. Find T $\cap$ V.

b. Suppose you are searching on the Internet for information about Republican candidates in Oregon running for either the House of Representatives or the Senate.

i. Represent this search using Boolean operators.

ii. Represent this search using sets.

c. State whether each statement is true or false. If it is true, draw a diagram that illustrates the statement. If it is false, give a counterexample.

i. If A $\subset$ B, then A $\cap$ B = A.

ii. If 3 $\in$ S, then {3} $\subseteq$ S.

iii. If M $\cap$ N = $\varnothing$, then M - N = M.

iv. (A $\cap$ B)$^{\prime}$ = A$^{\prime} \cap$ B$^{\prime}$

Consider the code in the earlier exercise, along with maximum likelihood decoding.

a. Explain why any error in a single bit can be corrected using this code.

b. Can this code correct double errors? For example, suppose the message 111 is received as 001. That is, a double error occurred in transmission: both of the first two bits were changed. Using maximum likelihood decoding, will this error be corrected? Explain.

Many Web sites have a "help" page or an "advanced search" page that will help you search more efficiently. For example, consider the help page adapted from the Library of Congress Web site in 2014.

Based on the information given in the table:

a. describe in words the effect of the operator AND.

b. describe the effect of OR.

c. describe the effect of NOT.

A standardized identification number system is used for books around the world. Each book is assigned a unique code-an International Standard Book Number (ISBN). These codes allow books to be more accurately identified and more effectively organized, especially with computerized recordkeeping. From 1970 to 2007, ISBNs had 10 digits. Since January 1, 2007, ISBNs contain 13 digits, so the numbering system is sometimes called ISBN-13. In this task and in Extensions Task 14, you will consider the older ISBN system. See Extensions Task 15 for an analysis of the ISBN-13 system.

Consider ISBN 0-380-00832-7. A 0 or 1 in the first position indicates that a book was published in an English-speaking country. The next block of digits identifies the publisher. The third block is assigned by the publisher from among the block of numbers the Library of Congress gives the publisher and it identifies the specific book. These first three blocks of digits can be of different lengths, depending on the publisher of the book. The last digit is a check digit. Here is how the check digit works.

There are ten digits in an ISBN: abcdefghij.

Compute 10a + 9b + 8c + 7d + 6e + 5f + 4g + 3h + 2i + j.

If the number is valid, this sum will be equivalent to 0 mod 11.

a. Verify that the ISBN given above is valid.

b. Consider this code: 0-88385-423-0. Is this a valid ISBN? Enter the number into the Search feature of a bookseller's Web site. What happens? Calculate to see if the ISBN is valid.

Rewrite each expression in standard polynomial form.

a. 3x(4 - 2x) + 8(x$^2$ - 5x)

b. (x - 5)(4x$^2$ + 7x - 9)

c. (5x$^3$ - 7x$^2$ + 5) - (x$^3$ +10x - 12)

d. $\frac{4 x^3-6 x^2+2 x}{2 x}$

Solve each equation.

a. 2t + 4(5t + 9) = 3

b. 8y(y - 2) + 4(6y) = 4(2y + 1.5y + y)

c. -0.15(p + 3) = p(0.75p -1.25) - 0.75p$^2$

d. $\frac{3}{8} x-\frac{1}{4}\left(\frac{1}{2} x+4\right)=\frac{3}{2}$