Show that any open subset of R (w std. topology) is a countable union of open intervals.

What is the objective of this problem and enough to show ?

Minimum 320 words and unique content please. In your opinion, what is the most important aspect of starting a business? What passion or interests led you to the decision to create your business?
Thank you!

Show that, for each natural number n, (x − 1)(x − 2)(x − 3). . .(x − n) − 1 is irreducible over Q, the rationals.

Using the SATGPA data set in Stat2Data package. Test by using α= .05.

1) Create the following three variables and then print out all the six variables.

Create a new variable “SAT”, which is the sum of MathSAT and VerbalSAT.

Create second new variable “SATLevel”, and assign the value of “SATLevel” as 1 when

SAT<=1100, 2 when 1100<SAT<=1200, 3 when 1200<SAT<=1300, and 4 when

SAT>1300.

Create third new variable “GPALevel” and assign the value of “GPALevel” as 1 when

GPA<=2.8, 2 when 2.8<GPA<=3.3, 3 when 3.3<GPA<=3.5, and 4 when GPA>3.5

Print out all the data in the descending order of their GPALevel and the ascending order of

their SAT when GPALevel is the same.

2) Use the Chi-Square test to conclude if the SATLevel and GPALevel are independent.
3) Compute the mean and variance of “GPA” for each level of “GPALevel”, and compute the

correlation matrices for the four variables: MathSAT, VerbalSAT, GPA and SAT.
4) Do the data provide sufficient evidence to indicate that the mean of MathSAT is significantly greater

than the mean of VerbalSAT.
5) Test if the proportion of MathSAT greater than VerbalSAT is 0.6.

(2) PQRS is a quadrilateral and M is the midpoint of PS.
PQ = a, QR = b and SQ = a – 2b.
(a) Show that PS = 2b.
Answer(a)
[1]
(b) Write down the mathematical name for the quadrilateral PQRM, giving reasons for your answer.
Answer(b) .............................................................. because ...............................................................
............................................................................................................................................................. [2]
__________

A tram leaves a station and accelerates for 2 minutes until it reaches a speed of 12 metres per second.
It continues at this speed for 1 minute.
It then decelerates for 3 minutes until it stops at the next station.
The diagram shows the speed-time graph for this journey.
Calculate the distance, in metres, between the two stations

Carry out a numerical experiment to compare the accu- racy of Formulas (5) and (19) on a function f whose derivative can be computed precisely. Take a sequence of valuesforh,suchas4−n with0≦n≦12.

hint: For CE 4.3.4, use f(x) = sin x and x = 0.25, for example. Then the exact derivative is cos 0.25, so you can compare your results to it, compute errors, and study how they behave as h decreases. You may want to format your outputs so that for each n you print, on the same row, the values of h, approximate f 0 (0.25), error of the approximation, and the theoretically estimated error term (h 2/6 for formula (5) and h 4/30 for formula (19)—here we are using the fact that the derivatives of sin x are at most 1 in absolute value). Don’t forget to discuss your findings!

f′(x)≈ 1 [f(x+h)− f(x−h)] (5)

f′(x)≈ 1 [f(x+h)− f(x−h)] /2h − 1 /12h(f(x +2h)−2[f(x +h)− f(x −h)]− f(x −2h)) (19)

Describe in detail a Queueing Theory. Include in your discussion general examples of particular industries that might employ Queueing Theory and why? Finally, present a specific example showing in detail where Queueing Theory can be employed and show a stepwise solution using this theory.

Logic/ Game theoryLet f(n) count the different perfect covers of a 2-by-n chessboard by dominoes. Evaluate f(1),f(2),f(3),f(4), and f(5). Try and find (and verify) a simple relation that the counting function f satisfies. Compute f(12) using the relation.


Here is a solution it is titled exercise 4a.) in the packet on page 3:
http://jade-cheng.com/uh/coursework/math-475/homework-01.pdf  
Not sure on how to follow the logic.

Use models to show that each of the following statements is independent of the axioms of incidence geometry:

(a) Given any line, there are at least two distinct points that do not lie on it.

(b) Given any point, there are at least three distinct lines that contain it.

(c) Given any two distinct points, there is at least one line that does not contain either of them.

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c.

(B) Let p ≥ 2. Prove that if 2^p−1 is prime, then p must also be prime.

(Abstract Algebra)

"Primal"
MAXIMIZE  Z = 12X1 + 18X2 +10X3
S.T.
                             2X1  + 3X2   + 4X3    <= 50
                              -X1  +  X2     +  X3    <= 0
                              0X1  -  X2 + 1.5X3    <= 0
                                 X1, X2, X3 >=0

1.  Write the "Dual" of this problem.
2.  Write the "Dual of the Dual" of this problem.
For steps 3 & 4, use the Generic Linear Programming spreadsheet or the software of your choice.  
(Submit a file pdf, text, xls file, etc., indicating the solution values and objective function value.)
3.  Solve the "Primal" problem.  (Find values of X1, X2, X3 and the Maximum Z)
4.  Solve the "Dual" problem. (Find values of Y1, Y2, Y3 and the Minimum Z*)

Let {z0, z1, z2, . . . , z2017} be the solution set of the equation z^2018 = 1. Show that z0 +z1 +z2 +···+z2017 = 0.