Algebraists have proven (using more advanced techniques than ones we’ve discussed) that An is a simple group for n ≥ 5.

Using this fact, prove that for n ≥ 5, An has no subgroup of order n!/4 .

c) Prove that Sn is not simple for n ≥ 3.

d) Is GL(2, R) a simple group? Prove your claim.

1. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE?
2. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE that start or end with vowels?
3. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE that contain the substring LITERAL?
4. (5) Taking into account identical letters, how many ways are there to arrange the letters in the word ANTIDISESTABLISHMENTARIANISM?

8. Definition: A set A is finite if there exists a non-negative integer c such that there exists a bijection from A to {n ∈ N : n ≤ c}. (The integer c is called the cardinality of A.)

(a) Let A be a finite set, and let B be a subset of A. Prove that B is finite. (Hint: induction on |A|. Note that our proof can’t use induction on |B|, or indeed refer to “the number of elements in B” at all, because we don’t yet know that B is finite!)

(b) Prove that the union of two disjoint finite sets is finite.

(c) Prove that the union of any two finite sets is finite. (Hint: A ∪ B = A ∪ (B − A))

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R.

(a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P.

(b) Let R be the set of equivalence classes of P and let F : R → P be the derivative operator defined as F([f]) = df/dx. Is F well defined (i.e. is it a function)? Is it surjective? Is it injective?

Let A = {a1, a2, a3, . . . , an} be a nonempty set of n distinct natural numbers. Prove that there exists a nonempty subset of A for which the sum of its elements is divisible by n.

Solve the given initial-value problem. dx/dt = y − 1

dy/dt = −6x + 2y

x(0) = 0, y(0) = 0

1) a) Prove that the union of two countable sets is countable.

b) Prove that the union of a finite collection of countable sets is countable.

Sam got a car financing where he needs to do 48 monthly payments of $5,448.75 each, starting the moment he receives the vehicle. Given an annual interest rate of 18.4% compounded monthly:

(a) Calculate the spot price of the car.

(b) Elaborate the amortization schedule.

This problem is a complex financial problem that requires several skills, perhaps some from previous sections. Clark and Lana take a 30-year home mortgage of $124,000 at 7.4%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1200 per month. (a) Find their regular monthly payment. (Round your answer to the nearest cent.) $ (b) Find the unpaid balance when they begin paying the $1200. (Round your answer to the nearest cent.) $ (c) How many payments of $1200 will it take to pay off the loan? Give the answer correct to two decimal places. monthly payments (d) Use your answer to part (c) to find how much interest they save by paying the loan this way. (Round your answer to the nearest cent.) $

When Maria Acosta bought a car
2
1
2
years ago,
she borrowed $13,000 for 48 months at 7.8% compounded monthly. Her monthly payments are $316.15, but she'd like to pay off the loan early. How much will she owe just after her payment at the
2
1
2
-year
mark? (Round your answer to the nearest cent.)
$

A man has ​$380000 invested in three rental properties.One property earns

7.5​% per year on the​ investment, the second earns 8​% and the third earns 9​%.

The total annual earnings from the three properties is $ 32,000 and the amount invested at 9​%

equals the sum of the first two investments. Let x equal the investment at

7.5​%, y equal the investment at 8​%,and z represent the investment at 9​%.

A.Solve the system of equations to find how much is invested in each property.

Explain how you can solve the following problems using the QR factorization.

(a) Find the vector x that minimizes ||Ax − b1||^2 + ||Ax − b2||^2 . The problem data are the m × n matrix A and two m-vectors b1 and b2. The matrix A has linearly independent columns. If you know several methods, give the most efficient one.

(b) Find x1 and x2 that minimize ||Ax1 − b1||^2 + ||Ax2 − b2||^2 . The problem data are the m × n matrix A, and the m-vectors b1 and b2. The matrix A has linearly independent columns.

Consider the following second-order ODE,

y"+1/4y=0

with y(0) = 1 and y'(0)=0.

Transform this unique equation into a system of two 1st-order ODEs.

Solve the obtained system for t in [0,0.6] with h = 0.2 by MATLAB using Euler Method or Improved Euler Method.

Use Laplace transforms to solve the initial value problem

x′′+4x′+3x=t,x(0)=0,x′(0)=3,

where primes indicate derivatives with respect to t.

Solve y''-y'-2y=e^t using variation of parameters.