5.

(a) Let σ = (1 2 3 4 5 6) in S6. Show that G = {ε, σ, σ^2, σ^3, σ^4, σ^5} is a group using the operation of S6. Is G abelian? How many elements τ of G satisfy τ^2 = ε? τ^3 = ε? ε is the identity permutation.

(b) Show that (1 2) is not a product of 3-cycles. Must be written as a proof!

(c) If a^4 = 1 and ab = b(a^2) in a group, show that a = 1. Must be written as a proof!

(d) Show that a group G is abelian if and only if (gh)^2 = (g^2)(h^2) for all g and h in G. Must be written as a proof!

Show that
(a)Sn=<(1 2),(1 3),……(1 n)>.
(b)Sn=<(1 2),(2 3),……(n-1 n)>
(c)Sn=<(1 2),(1 2 …… n-1 n)>

Supply proofs for the following miscellaneous propositions from the course in a metric space context:

(a) A convergent sequence is bounded.

(b) The limit of a sequence is unique.

(c) A n -neighborhood is an open set.

(d) A finite union of open sets is open.

(e) A set is open if and only if its complement is closed.

(f) A compact set (you may use either definition) is closed and bounded.

below are EXCEL outputs for various estimated autoregressive models for Coca-Cola's real operating revenues (in billions of dollars) from 1975 to 1998. From the data, we also know that the real operating revenues for 1996, 1997, and 1998 are 11.7909, 11.7757 and, 11.5537, respectively.                                                                                                                                                                                                                         AR(1) Model:

AR(2) Model:

AR(3) Model:

Referring to Table 16-4 and using a 5% level of significance, what is the model that uses the most lag variables?

Question 7 options:

how to represent the following in a linear model y=Xb + e where b' = (u,a1,a2,a3,y1,y2,y3) and e'=(e11,e12,e13,e21,e22,e23,e31,e32,e33)

' means transpose

y11=u+a1+y1+e11

y12=u+a1+y2+e12

y13=u+a1+y3+e13

y21=u+a2+y1+e21

y22=u+a2+y2+e22

y23=u+a2+y3+e23

y31=u+a3+y1+e31

y32=u+a3+y2+e32

y33+u+a3+y3+e33

X is a matrix pattern but I am confused about 1 and 0 placement

Also can X'X and X'y be determined and written in normal equations

The ‘Exclusive OR’ operation (also called XOR) between two propositions p and q is defined as follows:

p ⊕ q = (p ∨ q) ∧ ¬(p ∧ q)

Using laws of propositional logic prove that conjunction distributes over Exclusive OR, i.e, for any proposition r,

r ∧ (p ⊕ q) ≡ (r ∧ p) ⊕ (r ∧ q).

Clearly state which law you are using in each step.

Use Laplace transforms to solve the following initial value problem :

y^'' - 3y^' +2y = 1 + cos (t) + e^t , y(0) =1, y' (0) = 0

bit of a math problem. so if I retire at age 65 and withdraw 50,000 dollars from my 401k every year until I die at age 72, and there is an inflation rate of 2%, how much money do I need to initially be in my 401k to make it to my death without going broke?

Having trouble thinking of what formula to use because compound interest using the inflation rate is one thing but I'm not sure what to do about the 50,000 plus whatever interest that i would keep dishing out each year.

Let A be a closed subset of a
T4
topological space. Show that A with the
relative topology is a normal T1

Solve the differential equation by variation of parameters.

y'' + 3y' + 2y = cos(e^x)

y(x) = _____.

Determine if the binary relation <= is a partial order on A in the following cases:

(a) A = N × N and (a1, b1) (a2, b2) ⇔ a1 <= a2 for (a1, b1),(a2, b2) ∈ A

(b) X = {1, 2, 3, 4}, A = P(X) and a <= b ⇔ #a <= #b for a, b ∈ A (Here #a denotes the number of elements in the set a)

(c) A = N and a <= b ⇔ ∃k ∈ N : a^k = b

Find the general solution of the differential equation y′′+16y=17sec2(4t) 0<t<pi/8

Problem A: A toy maker has enough wood to make 24 small toys or 6 large ones. In other words, the large toys require 4 times the amount of wood as the small toys. She has Time to make 16 of the Small toys or 8 of the Large toys. The Small toys sell for $15 each, while the Large toys bring $45 each. Find the combination of Small and Large toys that maximizes her Revenue dollars.

Let S= number of Small toys, L = number of Large toys, R = Revenue from sales, W = Wood used, and T = Time used.

1. Formulate the problem:

a. Explain in words what are your Decision Variables?

b. Formulate the Objective Function.

c. Formulate the Constraints.

2. Graph the constraints and show the Feasible Region.

b. Graph the Isorevenue line and show the solution Graphically.

c. State the coordinates of all the Corners Points

d. Find the solution by Corner Point Solution method.

develop the recursion relation for the series solution to Bessel’s Equation x2y'' + xy' + (x2 - n2)y = 0. This solution y(x), sometimes called the “regular solution”, is the Bessel Function of the First Kind, Jn(x). Provide the appropriate mathematical background (solutions) that generates the Bessel Functions of the First Kind. Take the constant a0 = 1 for normalization purposes. Generate the solutions corresponding the n = 0, 1, and 2, and plot them over the range x = 0 to 3π, overlaying the graphs of the three functions and labeling them appropriately.