Prove using the short north-east diagonals or any other mathematical method of your preference, that if A is enumerable, then it is also countable with an enumeration that lists each of its members exactly three (3) times. Hint. Your proof will consist of constructing an enumeration with the stated requirement.

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Given a group G with a subgroup H, define a binary relation on G by a ∼ b if and only if ba^(-1)∈ H.

(a) (5 points) Prove that ∼ is an equivalence relation.

(b) (5 points) For each a ∈ G denote by [a] the equivalence class of a and prove that [a] = Ha = {ha | h ∈ H}. A set of the form Ha, for some a ∈ G, is called a right coset of H in G.

(c) (5 points) Let a ∈ G. For all g ∈ G prove that Hg = Ha if and only if g ∈ Ha. Hint: two elements are equivalent if and only if their equivalence classes coincide.

(d) (5 points) Prove that the map ρa : H → Ha given by ρa(h) = ha, h ∈ H, is a bijection.

find the equation of the line that has slope - 2/3 and which passes through (-1,-6)

The SkyLight Company produces two light fixtures (products 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product 1, two unit of frame parts and two units of electrical components are required. For each unit of product 2, three units of frame parts and two unit of electrical components are required. The company has 240 units of frame parts and 200 units of electrical components. Each unit of product 1 gives a profit of $1, and each unit of product 2 gives a profit of $3. No more than 60 units of product 2 should be produced.

1. Formulate this linear programming model algebraically.
2. Use the graphical method (by drawing the objective function line) to solve this model, determine the optimal solution and maximum profit.
3. Write down the coordinate of all corner points.
4. How does the optimal solution change if the maximum unit of product 2 is increased to 80 units? Use the graphical method to solve the new model.
5. Formulate the model in (a) using Excel spreadsheet. Identify the data cells, the changing cells, and the objective cell.
6. Solve the model using Excel Solver. Write down the optimal solution, and value of slack variables of each constraint.

3. Consider the volume E as the solid tetrahedron with vertices (1, a, 0), (0, 0, 0), (1, 0, 0), and (1, 0, 1) where a > 0. (a) Write down the region E as a type I solid. (b) Find a such that RRR E x^2 yz dV = 1.

: For each conjecture below, you are to describe in words what the Null Hypothesis and Alternative Hypothesis are. Consider the decision that you have to make based upon your conjectures. Explain in words or with a chart what the Type I and Type II errors mean in context. Finally, describe the ramifications of making these errors within the context of the problem and describe which of the 2 errors are worse (in your opinion).

1. It is believed that a new drug can cure a cold.
2. The teacher will never check homework today.
3. In a big store like Wal-Mart, they will never catch me if I shoplift.
4. If I try to learn to ski, I will end up hurting myself.

Determine the matrices associated with ProjΠ and ReflΠ where Π : 2x + 5y – z = 0

Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x − ∂M/∂y)/(M − N)
is a function of x + y. That is, let f(t) be a function such that

f(x + y) = (∂N/∂x − ∂M/∂y)/(M − N)
Assume that you can solve the differential equation
M dx + N dy = 0

by multiplying by an integrating factor μ that makes it exact and that it can also be
written as a function of x + y, μ = g(x + y) for some function g(t). Give a method
for finding this integrating factor μ, and use it to find the general solution to the
differential equation

(3 + y + xy)dx + (3 + x + xy)dy = 0.

Please answer all parts of the question. Please show all work and all steps.

1a.) Show that the solutions of x' = arc tan (x) + t cannot have maxima

1b.) Find the value of a such that the existence and uniqueness theorem applies to the ivp x' = (3/2)((|x|)^(1/3)), x(0) = a.

1c.) Find the limits, as t approaches both positive infinity and negative infinity, of the solution Φ(t) of the ivp x' = (x+2)(1-x^4), x(0) = 0

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Solve the initial value problem

dy/dx + H(x)y = e^5x ; y(0) = 2

where

H(x) = −1 0 ≤ x ≤ 3
1 3 < x

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold:

(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).

(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).

(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).

(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}