Prove that the real numbers do not have cardinality N₀ using Cantor’s diagonalization argument.

Let D₁₀ denote the dihedral group of the hexagon. Thus, D₁₀ is generated by r and f with r¹⁰=f²=1 and fr=r^-1f=r⁹f

(a) Show that D10 has a subgroups N and M such that
i. N ∼= D5 (isomorphic to D5)
ii. M is a cyclic subgroup group of order 2
iii. N ∩ M = {e}
iv. N and M are each normal in D10
v. Every element in g ∈ D10 is a product g = nm of elements n ∈ N and m ∈ M.

(b) Prove that if G is any group with normal subgroups N and M such that N ∩ M = {e}, then
nm = mn for all n ∈ N and m ∈ M.

Do a case study of Northeastern Airlines.

Northeastern Airlines is a regional airline serving nine cities in the New England states as well as cities in New York, New Jersey, and Pennsylvania. While nonstop flights are available for some of the routes, connecting flights are often necessary. Northeastern Airlines Service Area The network shows the cities served and profit in U.S. dollars per passenger along each of these routes. The routes from ?Boston-to-Providence and from Providence-to-Boston make only $ 9 per passenger profit after all expenses. To service these cities, Northeastern operates a fleet of sixteen 122-passenger Embraer E-195 jets. These jets, which were first introduced by Embraer in late 2004, have helped Northeastern Airlines remain profitable for a number of years. However, in recent years, the profit margins have been falling, and Northeastern is facing the prospect of downsizing their operations. Management at Northeastern Airlines has considered several options to reduce cost and increase profitability. Due to Federal Aviation Administration regulations, the company must continue to serve each of the nine cities. How they serve these cities, however, is up to the management at Northeastern. One suggestion has been made to provide fewer direct flights, which would mean that a city served by Northeastern might only have direct flights to one other city. The company plans to hire a marketing analytics consultant to determine how demand would be impacted by longer flights with more connections, and to forecast the demand along each of the routes based on a modified flight operations map. Before hiring the consultant, the company would like to first determine the most profitable (on a profit per passenger basis) way to continue serving all of the cities.

1. Find a Cartesian equation for the curve.

r cos(θ) = 2

Identify the curve.

2. Find a Cartesian equation for the curve.

r = 4 sin(θ)

Identify the curve.

In a certain country license plates consist of zero or one digit followed by four or five uppercase letters from the Roman alphabet.

(a)

How many different license plates can the country produce?

(b)

How many license plates have no repeated letter?

(c)

How many license plates have at least one repeated letter?

(d)

What is the probability that a license plate has a repeated letter? (Round your answer to the nearest whole percent.)

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The Cantor set, C, is the set of real numbers r for which Tⁿ(r) ϵ [0,1] for all n, where T is the tent transformation. If we set C₀= [0,1], then we can recursively define a sequence of sets C_i, each of which is a union of 2ⁱ intervals of length 3^-i as follows: C_i+1 is obtained from C_i by removing the (open) middle third from each interval in C_i. We then can define the Cantor set by

C= i=0 to infinity C_i
In general, a set S is called self-similar if for some real number r the scale of S by r can be exactly covered (without overlap) by a finite number, say n, of copies of the original set S. Then if r^d=n we say that d is the similarity dimension of the set S.
1. Consider the Cantor set as described above.
a. What is the length of the Cantor set?
b. Find the similarity dimension of the Cantor set.

1. Rachael runs 2 km to her bus stop, and then rides 4.5 km to school. On average, the bus is 45 km/h faster than Rachael’s average running speed. If the entire trip takes 25 min, how fast does Rachael run?

2. Write an equation of a rational function that satisfies all of these conditions

● Vertical asymptote at x = -8 and x = 5

● Horizontal asymptote at y = 0

● x-intercept at (-2, 0)

● f(0) = -2

● has a hole at x = 3

1) Define a sequence of polynomials H n (x ) by H 0 (x )=1, H 1 (x )=2 x , and for n>1 by H n+1 (x )=2 x H n (x )−2 n H n−1 (x ) . These polynomials are called Hermite polynomials of degree n. Calculate the first 7 Hermite polynomials of degree less than 7. You can check your results by comparing them to the list of Hermite polynomials on wikipedia (physicist's Hermite polynomials).

2) Use the power series method to solve the differential equation y ' '−2 x y '+λ y=0 where λ is an arbitrary constant. Verify that you get two independent solution y1, y2 by choosing a0=1, a1=0 and a0=0 , a1=1 . Show that the series expansion for one of the two solutions will terminate resulting in a polynomial solution when λ is chosen to be a positive even integer, λ=2 ,4,6,8,10 ,12 ,14 ,.... Rescale the polynomial solution so it starts with 2 n x n + lower powers of x , n=λ/2. Calculate the list of polynomials obtained this way and compare them to your solution of problem 1)

How many distinct 2x2 matrices can we have by using the numbers 1,2,3 and 4. Repeating numbers is not allowed.

2!*2!

4*3*4*3

4!

How many multiplications are required to compute AB if A is a 4x7 matrix and B is a 7x5 matrix?

35

28

980

140

4^4

Let A be an nxn matrix, What is det(A) if A has a row of zeros?

n-1

1

0

n

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample.

Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation.

A) If φ is injective, then dim(V) ≤ dim(W).

B) If dim(V) ≤ dim(W), then φ is injective.

C) If φ is surjective, then dim(V) ≥ dim(W).

D) If dim(V) ≥ dim(W), then φ is surjective.

E) If V = {0} , then φ is injective.

F) If dim(V) NOT= dim(W), then φ is not bijective.

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a) How many arrangements of all the letters in AABBCCD starts with A but does not end with A?

b) Find the number of arrangements of all the letters in AABBCCD in which none of the patterns AA, BB or CC occurs.

How many ways are there to rearrange the letters in MARKER?

Show Work

Find the closed formula solution to each of the following recurrence relations with the given initial conditions. Use an iterative approach and show your work! What is a_100? a) a_n=a_(n-1)+2,a_0=3 b) a_n=a_(n-1)+2n+3,a_0=4 c) a_n=2a_(n-1)-1,a_0=1 d) a_n=-a_(n-1),a_0=5

There are three vectors in R4 that are linearly independent but not orthogonal: u = (3, -1, 2, 4), v = (-2, 7, 3, 1), and w = (-3, 2, 4, 11). Let W = span {u, v, w}. In addition, vector b = (2, 1, 5, 4) is not in the span of the vectors. Compute the orthogonal projection bˆ of b onto the subspace W in two ways: (1) using the basis {u, v, w} for W, and (2) using an orthogonal basis {u' , v' , w'} obtained from {u, v, w} via the Gram Schmidt process. Finally, explain in a few words why the two answers differ, and explain why only ONE answer is correct.