In each part of this exercise, find a set A that makes the sentence true or explain why no solution can be found. a. $\emptyset \subseteq A$ b. $\emptyset \in A$ c. $A\subseteq \emptyset$ d. $A\in\emptyset$

Josea went out to dinner at an Indian restaurant. The total bill was $38. She wanted to leave a 15% tip. a. If you use the idea of scaling to find the tip amount, what would she need to multiply by? As you talk about this with your team, consider: How could you represent this multiplier as a fraction? How could you represent it as a decimal? Does it make a difference which representation, fraction or decimal, you use to solve this problem? Which do you think will be easier? b. How much should Josea leave for the tip? Show your calculations. c. If Josea changes her mind and wants to leave a 20% tip instead, how much will this be?

The vertices of a triangle are

$$A ( 0 , - 3 ) , B ( 2 , - 1 ) , \text { and } C ( 3 , - 3 )$$

. You translate the triangle 5 units right and 2 units down. Then you translate the image 3 units left and 8 units down. Is the original triangle congruent to the final image? If so, give two ways to show that they are congruent.

Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions.

$$\left. \begin{array} { l } { 3 x - 4 y = 12 } \\ { y = \frac { 3 } { 4 } x - 3 } \end{array} \right.$$

The following table indicates the number of unpaired electrons in the crystal field diagrams for some complexes. Complete the table by classifying each species as weak field, strong field, or insufficient information.

$$\begin{matrix} & \text{Unpaired}\\ \text{Species} & \text{Electrons} & \text{Classification}\\ \hline \text{$\mathrm { Fe } ( \mathrm { CNS } ) _ { 6 } ^ { 4 - }$} & \text{0} & \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{$\mathrm { CoCl } _ { 4 } ^ { 2- }$} & \text{3} & \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{$\mathrm { Fe } \left( \mathrm { H } _ { 2 } \mathrm { O } \right) _ { 6 } ^ { 3 + }$} & \text{5} & \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{$\mathrm { Fe } ( \mathrm { CN } ) _ { 6 } ^ { 4 - }$} & \text{0} & \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ \end{matrix}$$

(a) What injury does Beiler suffer in the wilderness? (b) What details make Beiler's pain come alive for the reader?

$$\begin{array} { l } { \text { Given that } \int _ { 0 } ^ { 1 } 3 x \sqrt { x ^ { 2 } + 4 } d x = 5 \sqrt { 5 } - 8 , \text { what is } } \\ { \int _ { 1 } ^ { 0 } 3 u \sqrt { u ^ { 2 } + 4 } d u ? } \end{array}$$

Which of the following equations do not represent linear functions? Explain. A.

$$12 = 2 x ^ { 2 } + 4 y ^ { 2 }$$

B.

$$y - x + 3 = x$$

C.

$$x = 8$$

D.

$$x = 9 - \frac { 3 } { 4 } y$$

E.

$$y = \frac { 5 x } { 11 }$$

F.

$$y = \sqrt { x } + 3$$

Complete each of the following by writing either $\in$ or $\subseteq$ in place of the $\bigcirc$. a. 2$\bigcirc${1,2,3}. b. {2}$\bigcirc${1,2,3}. c. {2}$\bigcirc${{1},{2},{3}}. d. $\emptyset\bigcirc${1,2,3}. e. $\mathbb{N}\bigcirc\mathbb{Z}$. f. {2}$\bigcirc\mathbb{Z}$. g. {2}$\bigcirc 2^{\mathbb{Z}}$.

Decide whether the words in the following pair are synonyms or antonyms. foreboding/prediction

In transmitting a bit from location A to location B, if we let X denote the value of the bit sent at locationAand Y denote the value received at location B, then H(X)-

$$H_y$$

(X) is called the rate of transmission of information from A to B. The maximal rate rate of transmission, as a function of P{X=1}=1-P{X=0}, is called the channel capacity. Show that, for a binary symmetric channel with

$$P \{ Y = 1 | X = 1 \} = P \{ Y = 0 | X = 0 \} = p$$

, the channel capacity is attained by the rate of transmission of information when

$$P \{ X = 1 \} = \frac { 1 } { 2 }$$

and its value is 1+p log p+(1-p)log(1-p).

Complete this indirect argument that $\sqrt { 2 }$ is irrational. Assume that a. ____. Then $\sqrt { 2 } = \frac { p } { q }$, where p and q are positive integers that have no common factors. Thus 2=b. ____, and $p ^ { 2 } = \mathbf { c }$. ____. This implies that $p ^ { 2 } = \mathbf { c }$ is even, and thus p is even. Since $p^2$ is the square of an even number, $p^2$ is divisible by 4 because d. ____. But then $q^2$ must be even because e. ____, and so q is even. Then p and q have a common factor of 2, which contradicts the assumption that p and q have no common factors.

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for

$$a_n$$

to find

$$a_8$$

the eighth term of the sequence.

$$1,2,4,8 , \dots$$

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent

$$\left. \begin{array} { l } { 8 y - 3 x = 15 } \\ { - 16 y + 6 x = - 30 } \end{array} \right.$$