Let N be a normal subgroup of the group G.

(a) Show that every inner automorphism of G defines an automorphism of N.
(b) Give an example of a group G with a normal subgroup N and an automorphism of N that is not defined by an inner automorphism of G

Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2, v3 be two linearly independent eigenvectors of A corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2, v3 are linearly independent.

As the new owner of a supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to set the price so that your revenue from selling it is as large as possible. Previous sales figures of the cheese are shown in the following table.

Price per Pound, p $3.00 $4.00 $5.00

Monthly Sales, q (pounds) 402 284 222

(a) Use the sales figures for the prices $4 and $5 per pound to construct a demand function of the form q = Ae−bp, where A and b are constants you must determine. (Round A and b to two significant digits.) q =

(b) Use your demand function to find the price elasticity of demand at each of the prices listed. (Round your answers to two decimal places.) p = $3, E = p = $4, E = p = $5, E =

(c) At what price should you sell the cheese to maximize monthly revenue? (Round your answer to the nearest cent.) $

(d) If your total inventory of cheese amounts to only 200 pounds, and it will spoil one month from now, how should you price it to receive the greatest revenue? (Round your answer to the nearest cent.)

Fake news reporter Chocolate Medaughter has had another interesting scoop to tell to the world. Suppose that the rate of exposure of this new story to new people is proportional to the number of people who have not seen the story out of L possible viewers (limited growth). Suppose that initially, at time t = 0 days, no one has heard of the story, and it takes 10 days to expose the story to 10% of L. How long will it take to expose the story to 50% of L?

For the function f(x) = x^2 +3x / 2x^2 + 7x +3 find the following, and use it to graph the function.

Find: a)(2pts) Domain

b)(2pts) Intercepts

c)(2pts) Symmetry

d) (2pts) Asymptotes

e)(4pts) Intervals of Increase or decrease

f) (2pts) Local maximum and local minimum values

g)(4pts) Concavity and Points of inflection and

h)(2pts) Sketch the curve

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} = 0.4:

1 2 3 4 5 6

1 0 0 0 0 1 0

2 .2 .05 0 .6 0 .15

3 0 0 .8 0 0 .2

4 0 .6 0 .2 0 .2

5 1 0 0 0 0 0

6 0 0 .7 0 0 .3

a. What is P{X1 = 4, X2 = 6 | X0 = 2}?

b. What is P{X2 = 6 | X0 = 2}? What is P{X18 = 6 | X16 = 2}?

c. What is P{X0 = 2, X1 = 4, X2 = 6}?

d. What is P{X1 = 4, X2 = 6}?

Why is it not good to directly store the hash of password in a file? How can the use of salt enhance password security?

7. (10 pts.) Amelia visits several islands. The individuals on each these islands are either truth-tellers or liars. Amelia is interested in finding out whether everyone on a given island is a truth-teller, or whether everyone is liar, or whether there are both truth-tellers and liars. (Note: it only takes one liar for an island’s inhabitants to include both truth-tellers and liars.) Amelia is also interested in finding someone from whom to bum a cigarette. (Yes, she shouldn’t smoke, but she does anyway and intends to stop soon.) In each of the following questions, use complete English sentences to present the reasoning that backs up your answer. By the way, in this problem, and indeed every problem, you should remember that any universally generalized conditional whose antecedent-predicate is true of nothing is true. Amelia has already visited two of these islands. Now she visits two more.

a. On the third island, Amelia asks whether anyone on the island smokes. Everyone answers: “If I smoke, then everyone smokes.” What can Amelia conclude about the third island? b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

Question 2: Pocket Politics (any resemblance to recent events is purely coincidental) Long time ago, in a country far far away, a Totally Racist Unqualified Malicious President ruled the land with an iron fist. His rivals had to do everything in their power to free the people from his terrible regime and make the country great again. Many people offered to take the TRUMP down, but six wise and brave men and women stood out from the rest:
1. Joe “Busy Hands” Bye-then (J) 2. Bernie “Crazy Eyes” Slanders (B) 3. Elizabeth “Pocahontas” Warden (E) 4. Tulsi “Go Land Crabs!” Globbard (T) 5. Mike “Mini Me” Broomfield (M) 6. Pete “Father-of-Chickens” Boot-a-Judge (P)
In order for the good people to make the right choice, the candidates have to gather in the town hall for a night of sword fights and verbal altercations. You are in charge of organizing the grand event.

Q1. Since no clear ranking could be established at the previous fights, the candidates are now paired up tournament style and each pair fights it out until someone gives up. How many ways do you have to divide the six candidates into three pairs? (Hint: How many ways can you select one pair? How many ways does it leave you to select the second pair? Then the third? Don’t forget to eliminate redundancy due to order).

1. . Let A = {1,2,3,4,5}, B = {1,3,5,7,9}, and C = {2,6,10,14}.

a. Compute the following sets: A∪B, A∩B, B∪C, B∩C, A\B, B\A.

b. Compute the following sets: A∩(B∪C), (A∩B)∪(A∩C), A∪(B∩C), (A∪B)∩(A∪C).

c. Prove that A∪B = (A\B)∪(A∩B)∪(B\A).

2. Let C0 = {3n : n ∈ Z} = {...,−9,−6,−3,0,3,6,9,...} C1 = {3n+1 : n ∈ Z} = {...,−8,−5,−2,1,4,7,10,...} C2 = {3n+2 : n ∈ Z} = {...,−7,−4,−1,2,5,8,11,...}.

a. Prove that the sets C0, C1, and C2 are pairwise disjoint.

b. Compute C0 ∪C1 ∪C2.

3. Let R>0 be the set of positive real numbers, that is, R>0 = {x ∈ R : x > 0}.

Prove that {e x : x ∈ R} = R>0 and {logx : x ∈ R>0} = R.

SupposeG=〈a〉is a cyclic group of order 12.

Find all of the proper subgroups of G, and list their elements. Find all the generators of each subgroup. Explain your reasoning.

Give an example of a nonabelian group G of order n and a subgroup H of order k. Then list all of the cosets of G/H. where n = 24 and k = 3.

A bottle of

milkmilk

initially has a temperature of

7575degrees°F.

It is left to cool in a refrigerator that has a temperature of

4545degrees°F.

After 10 minutes the temperature of the

milkmilk

is

5757degrees°F.

a. Use​ Newton's Law of​ Cooling,

Upper T equals Upper C plus left parenthesis Upper T 0 minus Upper C right parenthesis e Superscript ktT=C+T0−Cekt​,

to find a model for the temperature of the

milkmilk​,

T​,

after t minutes.

b. What is the temperature of the

milkmilk

after 15​ minutes?

Tequals=nothingdegrees°F

​(Type an integer. Round to nearest​ degree.)

c. When will the temperature of the

milkmilk

be

5252degrees°​F?

tequals=nothing

minutes

A fishing boat leaves a dock at 2:00pm and travels due east at a speed of 20km/h. Another boat has been heading due north at 15 km/h and reaches the same dock at 3:00pm. At what time were the two boats closest together?