Mathematics - 12 December 2007

Let (r_n) be an enumeration of the rationals and let (a_n) be a sequence of positive numbers. For each n =1,2,3...define I_n as the open interval (r_n - a_n , r_n + a_n) and let I = Union (n = 1, oo) I_n. It's clear that I is dense in R.

Now, consider the series Sum (n=1, oo) a_n. We have 2 possibilities:

(1) If the series converges, then, according to the sub-additivity of the measure (Lebesgue measure, in this case), we have m(I) <= Sum (n=1, oo) m(I_n) = Sum (n=1, oo) (2 a_n) = 2Sum (n=1, oo) a_n < oo. Since m(R) = oo, it follows I is not all of R, but one of its proper subsets.

(2) If, on the other hand, Sum (n=1, oo) a_n diverges, then the argument given in (1) leads to no conclusion, all we get is the obvious inequality m(I) <= oo. In this case, is it possible that I is still a proper subset of R or will we always have I = R?

If lim inf a_n >0, then I = R, but if If lim inf a_n = 0, I'm not sure.

Thank you for any suggestion.

2007-12-12 06:58:23 · 2 answers · asked by Steiner 7

2007-12-12 06:57:30 · 3 answers · asked by william5000forever 2

so it creates the right graph. Is it: ln (x) / -2?

2007-12-12 06:52:13 · 6 answers · asked by Anonymous

Because a function never assigns two output values to a single input value, its graph never has
two different points with the same ___-coordinate.

I think its y-coordinate
Is that correct?

2007-12-12 06:25:05 · 2 answers · asked by Anonymous

Use the discriminate to determine the number of solutions for the equation 9x2 + 6x = -1.
1
2
3
none

2007-12-12 06:23:39 · 4 answers · asked by Anonymous

0.9(x+80)+0.1x=97

2007-12-12 06:19:55 · 2 answers · asked by higherlovetx 5

Two cities are1460 miles apart. A plane leaves one of them, traveling towards the other. At the same time a plane leaves the other city, traveling towards the first at an average speed of 380 miles per hour. How many hours will it take them to meet?

2007-12-12 06:08:21 · 4 answers · asked by higherlovetx 5

Use the function f(x) = 3x-2 to

1. Determine f^-1
2.If f(a) = 4, determine the numerical value of f^-1 x (4)

2007-12-12 05:56:58 · 2 answers · asked by Anonymous

2007-12-12 05:54:52 · 9 answers · asked by Kameron W 1

Find the perimeter of the triangle shown in the figure.

Note: The triangle shows the following numbers on each side...
top:4; left-square root 5(negative sign in the middle) and sqare root 3...right-square root 5(positive sign in the middle) and square root 3...PLEASE HELP

2007-12-12 05:42:54 · 5 answers · asked by God Bless America!~ 4

use the co-ordinates provided in the grid sheet and apply graphical techniques to set out the buildings such as plot 1

Co-ordinates
PLOT 1
Eastings Northings
2/000 12/000
5/100 9/500
6/800 16/700
8/900 14/300

2007-12-12 05:40:33 · 7 answers · asked by sabeen 2

5x+4y=6
3x-4y=10

so y=
x=

2007-12-12 05:39:44 · 7 answers · asked by churrrrrrrrrrrrn 1

in the account grows to 5304 dollars?

2007-12-12 05:38:55 · 6 answers · asked by simonkf2002 1

6x-2y=6

i'm trying to locate the x- and y-intercepts in ordered pairs. this is very confusing to me. Please help. Thank you.

2007-12-12 05:35:49 · 5 answers · asked by higherlovetx 5

2007-12-12 05:23:52 · 9 answers · asked by Anonymous

2007-12-12 05:21:49 · 5 answers · asked by Ben 1

2007-12-12 05:19:11 · 5 answers · asked by clove88 2

The speed of a stream is 4 mph. A boat travels 7 miles upstream in the same time it takes to travel 15 miles downstream. What is the speed of the boat still in the water?

2007-12-12 05:17:41 · 9 answers · asked by Caitlin 2

Thank You everyone for all the help.

Could someone walk me through how to solve this question. I know there will be many coming up on a test I have.

Miss bass wants a 79% solution of herbicide to be mixed with a solution of 62% herbicide to form 34 litersof 66% solution. How many liters of the 79% solution must she use?

Thank You so much

2007-12-12 05:09:59 · 2 answers · asked by Anonymous

im not in algebra but iv been learning on my own to solve small and easy algebra problems and in many problems on yahoo answers if found this sighn:^ i would like to know what it means
thank you

2007-12-12 05:06:54 · 11 answers · asked by Anonymous

I have started up a business of making box kites (see diagram included for “blue prints of a box kite). My
kites will be made of 4 “length” pieces and 8 “cross sectional pieces”.
Here are my conditions.
(1) The kite’s frame I build must be made from 6 meters of balsa wood, which costs me $4.00/m
(2) the kite’s “base” will be square
(3) I will cover the surfaces as shown on the diagram, but the red covering costs me $5.00/m2 while the
blue covering costs me $6.00/m2
I must design the kite such that I am aware of the total surface area of my kite as well as the cost of my
kite. So you will carry out 2 mathematical investigations to help me decide the optimal dimensions of the
kite so that I can (a) maximize my surface area and (ii) minimize my cost
PART A. You must decide what dimensions of the kite (length and width) will maximize the surface area
of the kite. Your solution will include (i) a clear and concise outline of what you have outlined, and (III) explain all your step

2007-12-12 05:02:15 · 2 answers · asked by Anonymous

far must one go to reduce the strength by 66%. (i.e., 34% of its original strength)?

2007-12-12 05:02:01 · 3 answers · asked by simonkf2002 1

minutes after it starts cooling is given in Celsius, by
H = 20 + 880e^-.1t
a. find the temperature of the bar at the end of one hour
b. Find the average value of the temperature over the first hour.

2007-12-12 04:58:03 · 1 answers · asked by simonkf2002 1

I solved it and got it down to this: x^2+y^2 = 2x^2

But I got a point taken off, I guess because it can be taken down one more step into: θ=pi/4

How do you make that step to that? How do you go from what I got to θ=pi/4 ? Explain please? Thanks!

2007-12-12 04:46:45 · 7 answers · asked by deb3267 1

I can't figure this out - I'm doing something wrong.

Instructions are to Divide and write my answer without exponents:

-9.8 x 10^-7 / 1.4 x 10^-4

Thank you so much for your assistance.

2007-12-12 04:42:21 · 7 answers · asked by higherlovetx 5

I have a take home test, and am stuck on this problem. I need to get this right, to pass. could someone help me solve this problem. it would be appreciated more then you know. Thank You.

An office manager has 5 cent stamps, eight cent stamps, and twenty cent stamps.
There are three times as many 20 cent stamps as five cent stamps. There are 2 fewer eight cent stamps then five cent stamps. The total value of the stamps in the collections is 2.76. Find then number of five cent stamps.

Cold anyone give me any help on solving this.

2007-12-12 04:38:19 · 9 answers · asked by Anonymous

I changed it to y^9 x^4/ x^6 y^2 but what do I do from there? I don't need to solve anything, just simplify it with positive exponents in the final answer.

Thanks for your help!!

2007-12-12 04:33:18 · 3 answers · asked by ♫ ∫aoli 4

2007-12-12 04:24:48 · 3 answers · asked by jason c 1

Abbey Ruckles went to the Hillside Bank. She borrowed 3,000 at a rate of 12 3/4%. The date of the loan was september 10. Abbey hoped to repay the loan on january 29. Assuming the loan is based on ordinary interest, how much will Abbey totally repay on January 29?

I did:

P=3000
R=12.75%
T=5/12
I=PxRxT
I=3000x12.75x141/360
I=3000x .1275x.3916
I=149.81
Answer is 3149.81?

2007-12-12 04:05:21 · 2 answers · asked by Jasmine 1

2007-12-12 04:01:28 · 5 answers · asked by Anonymous