Write the correct word in the space next to each definition.

random $\rule{2cm}{0.15mm}$

Let X={a, b, c}. Define a function S from $\mathcal{P}$(X) to the set of bit strings of length 3 as follows. Let Y$\subseteq$X. If a$\in$Y, set $s_1=0$; If a$\notin$Y, set $s_1=1$; If b$\in$Y, set $s_2=0$; If b$\notin$Y, set $s_2=1$; If c$\in$Y, set $s_3=0$; If c$\in$Y, set $s_3=1$. Define S(Y)=$s_1s_2s_3$.

Prove that S is onto.

Find the distance from the

(b). origin to the plane $2=z-x$.

Phoenix Industries has pulled off a miraculous recovery. Four years ago it was near bankruptcy. Today, it announced a $1 per share dividend to be paid a year from now, the first dividend since the crisis. Analysts expect dividends to increase by$1 a year for another 2 years. After the third year (in which dividends to increase by $1 a year for another 2 years. After the third year (in which dividends are$3 per share) dividend growth is expected to settle down to a more moderate long-term growth rate of 6%. If the firm's investors expect to earn a return of 14% on this stock, what must be its price?

The table shows the time it took several members of a cycling club to bike different distances. Sort the cyclists from least to greatest unit rate.

Integrative: Determining relevant cash flows Lombard Company is contemplating the purchase of a new high-speed widget grinder to replace the existing grinder. The existing grinder was purchased 2 years ago at an installed cost of $60,000; it was being depreciated under MACRS using a 5 -year recovery period. The existing grinder is expected to have a usable life of 5 more years. The new grinder costs$ 105,000 and requires $5,000 in installation costs; it has a 5-year usable life and would be depreciated under MACRS using a 5 -year recovery period. Lombard can currently sell the existing grinder for$ 70,000 without incurring any removal or cleanup costs. To support the increased business resulting from purchase of the new grinder, accounts receivable would increase by $40,000, inventories by$ 30,000, and accounts payable by $58,000. At the end of 5 years, the existing grinder would have a market value of zero; the new grinder would be sold to net$ 29,000 after removal and cleanup costs and before taxes. The firm is subject to a 40% tax rate. The estimated earnings before depreciation, interest, and taxes over the 5 years for both the new and the existing grinder are shown in the following table.

b. Determine the incremental operating cash flows associated with the proposed grinder replacement. (Note: Be sure to consider the depreciation in year 6.)

Explain the importance of microbes as agents of food-borne illnesses.

Find the complement of the following set.

The set {x|x is an integer strictly between 0 and 8} if U is the set of all integers.

(a) Recollect: At what time of year does the story happen? (b) Interpret: What is significant about the story taking place at this time?

In 1975, an airplane carrying an atomic clock flew back and forth for $15 \mathrm{~h}$ at an average speed of $140 \mathrm{~m} / \mathrm{s}$ as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. How much time did the airborne clock "lose" with respect to the clock on the ground?

Refer to the table. Which lap has the slowest time?

Suppose G is an infinite cyclic group with generator $\sigma$. (a) Prove that multiplication by

$$\sigma-1 \in \mathbb{Z} G$$

defines a free G-module resolution of

$$\mathbb{Z}: 0 \rightarrow \mathbb{Z} G \stackrel{\sigma-1}{\rightarrow} \mathbb{Z} G \rightarrow \mathbb{Z} \rightarrow 0.$$

(b) Show that

$$H^{0}(G, A) \cong A^{G},$$

that

$$H^{1}(G, A) \cong A /(\sigma-1) A.$$

and that $H^{n}(G, A)=0$ for all n $\geq$ 2. Deduce that

$$H^{1}(G, \mathbb{Z} G) \cong \mathbb{Z}$$

(so free modules neednot be cohomologically trivial).

The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle's orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect.

An RC series circuit consists of an AC source, a $33.0 - \mu \text F$ capacitor, and a 30.0-$\Omega$ resistor. If the source emf is given by $\varepsilon = \varepsilon_ { max } \sin ( \omega t )$, where $E _ { \max } = 180.0 \mathrm { V }$ and $\omega = 20.0 \mathrm { s } ^ { - 1 }$, what is the current amplitude?