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.

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.

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?

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.

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

In the diagram below, $\overleftrightarrow{A B} \| \overleftrightarrow{D E}$

a. Explain why $\triangle$ CED $\sim \triangle$ CBA.

b. Find the lengths of $\overline{A C}$ and $\overline{D E}$.

Rewrite each product as a single algebraic fraction. Then simplify the result as much as possible.

a. $\left(\frac{2 x^2}{9}\right)\left(\frac{12}{8 x}\right)$

b. $\left(\frac{3 a}{12}\right)\left(\frac{a+4}{a^2}\right)$

c. $\left(\frac{y^2-16}{y+1}\right)\left(\frac{y}{y+4}\right)$

Fifty randomly selected families in a neighborhood in a large city were surveyed and each family was asked how many vehicles were registered at that address. The results of the survey are summarized in the table below.

$$\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Vehicles } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Families } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Families } \end{array} \\ \hline 0 & 17 & \\ \hline 1 & 23 & \\ \hline 2 & 7 & \\ \hline 3 & 2 & \\ \hline 4 & 1 & \\ \hline \end{array}$$

a. Explain what the 23 in the table indicates.

b. Use the data in the table to compute the average number of vehicles registered per address in this neighborhood.

c. Determine the percentage of families in the neighborhood that had no vehicles registered at their address. Then fill in the rest of that column of the table with percentages. Enter each percentage as a decimal number between 0 and 1. The percentage of times that a data value occurs is also called the relative frequency.

d. You can also find the average number of vehicles per address using the relative frequencies. In the formula below, f(x) is the relative frequency of the data value x. Use this formula to find the average number of vehicles registered per address.

Mean $=\sum(f(x) \cdot x)$, where x takes on all possible data values.

Your answer here should match the one you got in Part b.

e. What is the median number of vehicles registered per address in this neighborhood?

f. Which measure of center is most appropriate for these data? Explain your reasoning.

Consider the following numbers:

$$\begin{array}{lllllllll} 123 & \frac{2}{3} & \pi & -\frac{8}{4} & \sqrt[3]{8} & 31.72 & -53 & 6 \frac{3}{8} & \sqrt{7} \end{array}$$

a. Which of the numbers are whole numbers?

b. Which of the numbers are integers?

c. Which of the numbers are rational numbers?

d. Which of the numbers is irrational numbers?

e. Order the numbers from smallest to greatest.

Write each expression in a simpler equivalent exponential form using only positive exponents.

a. (3x$^6$)$^2$

b. (3$x^6$)(5x$^2$)

c. $\frac{10 x^6}{4 x^2}$

d. $\frac{-2 x^2}{8 x^6}$

e. 14(7x)$^{-1}$

f. (3x$^{-6}$)(2x$^4$)

At the beginning of the school year, the 186 seniors at Sterling High School were surveyed and asked about whether they plan to go to college and if they had a summer job. Some of the results are shown in the chart below.

$$\begin{array}{|l|c|c|c|} \hline& \text { Had Summer Job } & \text { No Summer Job } & \text { Total } \\ \hline \text { College Plans } & & & \\ \hline \text { No College Plans } & 24 & & 72 \\ \hline \text { Total } & & 56 & 186 \\ \hline \end{array}$$

a. Fill in the remaining cells on a copy of the chart.

Suppose one of the seniors at Sterling High School is randomly chosen. Determine each of the following probabilities.

b. P(college plans)

c. P(had summer job)

d. P(college plans and had summer job)

e. P(college plans or had summer job)

f. What is the probability that someone who had college plans also had a summer job? (This can be written in symbols as P(had summer job | college plans). The symbol | means "given that" and restricts the group under consideration. So in this case, rather than determining the probability based on all of the students, we are only considering the 114 students who planned to go to college.)

g. P(college plans | had summer job)

In recent years, the number of students enrolled in colleges and universities in the United States has been increasing linearly. In 2001, there were approximately 15.9 million students enrolled in U.S. colleges and universities (including community colleges). In 2012, total U.S. college and university enrollment had increased to approximately 19.8 million students.

a. Using the information above, write a function rule that can be used to estimate student enrollment in U.S. colleges and universities. Let t represent the number of years since 2000 and E(t) represents the number of students in millions.

b. What is the slope of the graph of this function? Be sure to include units of measure in your answer. What does it tell you about student enrollment in U.S. colleges and universities?

c. What is the y-intercept of the graph of this function? What does it tell you about student enrollment in U.S. colleges and universities?

d. Use your function to estimate college and university enrollment in 2015.

e. Use your function to determine the year during which the enrollment will first be greater than 23 million.

A survey of 648 residents of Summit found that 480 of them thought the town should have a 4th of July parade.

a. What is the parameter that this survey was trying to estimate?

b. What is the point estimate of the proportion of residents of Summit who think the town should have a 4th of July parade?

c. Compute the margin of error for this survey. Then write a sentence explaining what it means in this context.