Analytic function on unit disk with power series has pole on unit circle, then power series diverges on unit circle.

Answers

Answer 1

Answer:

The function

{\ displaystyle f (z) = {\ frac {z} {1- | z | ^ {2}}}} {\ displaystyle f (z) = {\ frac {z} {1- | z | 2}

It is an example of real and bijective analytical function from the open drive disk to the Euclidean plane, its inverse is also an analytical function. Considered as a real two-dimensional analytical variety, the open drive disk is therefore isomorphic to the complete plane. In particular, the open drive disk is homeomorphic to the complete plan.

However, there is no bijective compliant application between the drive disk and the plane. Considered as the Riemann surface, the drive disk is therefore different from the complex plane.

There are bijective conforming applications between the open disk drive and the upper semiplane and therefore determined as Riemann surfaces, are isomorphic (in fact "biholomorphic" or "conformingly equivalent"). Much more in general, Riemann's theorem on applications states that the entire open set and simply connection of the complex plane that is different from the whole complex plane admits a bijective compliant application with the open drive disk. A bijective compliant application between the drive disk and the upper half plane is the Möbius transformation:

{\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}} {\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}}

which is the inverse of the transformation of Cayley.

Answer 2

if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

To understand why a power series diverges on the unit circle when the analytic function it represents has a pole on the unit circle, we can use the concept of analytic continuation and the properties of poles and singularities.

Here's a step-by-step explanation:

1.Analytic function on the unit disk: Let's consider an analytic function defined on the open unit disk, denoted by[tex]\(D = \{z \in \mathbb{C} : |z| < 1\}\)[/tex]. This means the function is holomorphic (complex differentiable) at every point within this disk.

2.Power series representation: Since the function is analytic on[tex]\(D\)[/tex], it can be represented by a power series expansion around any point [tex]\(z_0\) in \(D\)[/tex]. Let's denote this function by[tex]\(f(z)\)[/tex], and its power series representation centered at[tex]\(z_0\) by \(\sum_{n=0}^{\infty} a_n (z - z_0)^n\)[/tex].

3.Pole on the unit circle: Suppose[tex]\(f(z)\)[/tex] has a pole (a point where the function becomes unbounded) on the unit circle[tex]\(|z| = 1\)[/tex], i.e., there exists a point [tex]\(z_1\)[/tex] on the unit circle such that [tex]\(f(z_1)\)[/tex] is infinite. Without loss of generality, let's assume [tex]\(z_1 = 1\)[/tex] (since the unit circle is symmetric about the origin).

4.Behavior near the pole: Near the pole at [tex]\(z = 1\)[/tex], the function[tex]\(f(z)\)[/tex]can be expanded in a Laurent series, which includes negative powers of [tex]\((z - 1)\)[/tex]. This expansion will have infinitely many terms with negative powers, indicating the singularity at [tex]\(z = 1\)[/tex].

5.Radius of convergence: The radius of convergence of the power series representation of [tex]\(f(z)\)[/tex]is at least the distance from the center of convergence to the nearest singularity. In this case, since the singularity (pole) is on the unit circle, the radius of convergence of the power series cannot exceed 1.

6.Divergence on the unit circle: Since the radius of convergence of the power series representation of [tex]\(f(z)\)[/tex] is at most 1, the power series diverges at every point on the unit circle (except possibly at the point of singularity itself, where it may converge by definition). This divergence occurs because the function has a singularity (pole) on the unit circle.

Therefore, if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.


Related Questions

The measures of one acute angle in a right triangle is four times the measure of the other acute
angle. Write and solve a system of equations to find the measures of the acute angles.

Help with this exercise

Answers

Answer:

[tex]A_2[/tex] = 18°

[tex]A_3[/tex] = 4(18°) = 72°

Step-by-step explanation:

Given:

One angle in the triangle is 90°One angle that isn't 90° is 4 times larger than another angle that isn't 90°

Angles:

[tex]A_1[/tex] = 90°

[tex]A_2[/tex] = x

[tex]A_3[/tex] = 4x

Solution Pathway:

Under the rules for any triangle, a triangle's interior angles must add up to 180°. Using this, we can set up the equation:

sum of the interior angles = 180°90° + x + 4x = 180°

Now let's solve for x.

90 +x + 4x = 18090 + 5x = 1805x = 90x = 18°

Now that we know x is 18°, lets plug this value into the two unknown acute angles.

[tex]A_2[/tex] = 18°[tex]A_3[/tex] = 4(18°) = 72°

Answer:

72 degrees and 18 degrees.

Step-by-step explanation:

If the 2 angles are x and y, we have the system:

x + y = 90           (as it is a right triangle)

x = 4y                  (given).

Substitute x =- 4y in the first equation:

4y + y = 90

5y = 90

y = 18.

So  x + 18 = 90

x = 90 - 18

x = 72.

The scale model of a rectangular garden is 1.5 ft by 4 ft. The scale model is enlarged by a scale factor of 7 to create the actual garden. What is the area of the actual garden

Answers

Answer:

The Area of the actual garden is 294 square feet.

Step-by-step explanation:

The scale model of a rectangular garden is 1.5 ft by 4 ft.

Length of scale model=1.5 ft

Breath of scale model=4 ft

The scale model is enlarged by a scale factor of 7 to create the actual garden.

It means that the dimension of the garden are multiplied with the scale factor to find the actual dimension.

Hence,

Length of actual garden=[tex]1.5\times 7= 10.5\ ft[/tex]

Breath of actual garden=[tex]4\times 7 = 28\ ft[/tex]

Now Area of garden can be calculated by multiplying length and breadth.

Framing the equation we get;

Area of actual garden=[tex]10.5\ ft \times 28\ ft = 294\ ft^2[/tex]

Hence, area of the actual garden is 294 square feet.

Answer:294 took test

On July 31, Oscar checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What was the total amount of his check?

Answers

Answer:

$307.60

Step-by-step explanation:

multiple $76.90 by the 4 nights he stayed. $76.90×4

Answer:

76.00

Step-by-step explanation:

Which inequality does the given graph represent?
A) y > 3x + 4
B) y > 1/3x − 4
C) y > 1/3x + 4
D) y ≥ 1/3x + 4

Answers

Answer:

The answer to your question is letter C

Step-by-step explanation:

Process

1.- Find two points of the dotted line

   A (0, 4)

   B (3, 5)

2.- Find the slope of the line

   [tex]m = \frac{y2 - y1}{x2 - x1}[/tex]

Substitution

   [tex]m = \frac{5 - 4}{3 - 0}[/tex]

   [tex]m = \frac{1}{3}[/tex]

3.- Write the equation of the line

   y - y1 = m(x - x1)

   y - 4 = 1/3(x - 0)

   y - 4 = 1/3x

   y = 1/3x + 4

4.- Write the inequality

    We are interested on the upper area so

    y > 1/3x + 4

Answer: C) y > 1/3x + 4

Step-by-step explanation: The line is dashed hence >. The slope is 1/3 and the y- intercept is 4. This makes the inequality y > 1/3x + 4.

Hope this helps!  :)

A chain letter works as follows: One person sends a copy of the letter to five friends, each of whom sends a copy to five friends, each of whom sends a copy to five friends, and so forth. How many people will have received copies of the let- ter after the twentieth repetition of this process, assuming no person receives more than one copy?

Answers

Answer:

The number of  people that received copies of the letter at the twentieth stage is 9.537 × 10¹³ .

Step-by-step explanation:

Using the discrete model,

a_k = r a_(k-1) for all integers k ≥ 1 and a₀ = a

then,

aₙ = a rⁿ for all integers n ≥ 0

Let a_k be equal to the number of people who receive a copy of the chain letter at a stage k.

Initially, one person has the chain letter (which the person will send to five other people at stage 1). Thus,

a = a₀ = 1

The people who received he chain letter at stage (k - 1),  will send a letter to five people at stage k and thus per person at stage (k - 1), five people will receive the letter. Therefore,

a_k = 5 a_(k - 1)

Thus,

aₙ = a rⁿ = 1 · 5ⁿ = 5ⁿ

The number of  people that received copies of the letter at the twentieth stage is

a₂₀ = (5)²⁰ = 9.537 × 10¹³ copies

What additional information will allow you to prove the triangles congruent by the HL Theorem?A. A = EB. bce= 90C. ac = dcD. ac=bd

Answers

Answer:

C) AC=DC

Step-by-step explanation:

In the figure of question attached below:

In Δ ACB and  Δ DCE, AB and DE are hypotenuse respectively.

According to HL theorem:

"If hypotenuse and one leg of right angle triangle is congruent to Hypotenuse and one leg of other right angle triangle then triangles are congruent"

According to above statement if AB and DE are hypotenuse of Δ ACB and  Δ DCE respectively then either

AC = DC    (leg)

BC = EC     (leg)

In order to prove  congruence of triangles using HL theorem, AC must be equal to DC.

So option C is correct.

Answer:

C ) AC=DC

Step-by-step explanation:

edge 2021

£980 is divided between Caroline, Sarah & Gavyn so that Caroline gets twice as much as Sarah, and Sarah gets three times as much as Gavyn. How much does Sarah get?

Answers

Sarah has received £ 294

Solution:

Given that £980 is divided between Caroline, Sarah & Gavyn

Let "c" be the amount received by caroline

Let "s" be the amount received by sarah

Let "g" be the amount received by gavyn

Caroline gets twice as much as Sarah

amount received by caroline = twice as much as Sarah

amount received by caroline = 2(amount received by sarah)

c = 2s ---- eqn 1

Sarah gets three times as much as Gavyn

amount received by sarah = three times as much as Gavyn

amount received by sarah = 3(amount received by gavyn)

s = 3g ------- eqn 2

Given that total amount is 980

c + s + g = 980 --- eqn 3

Let us solve eqn 1, 2, 3 to get values of "c" "s" "g"

From eqn 2,

[tex]g = \frac{s}{3}[/tex]  --- eqn 4

Substitute eqn 1 and eqn 4 in eqn 3

[tex]2s + s + \frac{s}{3} = 980\\\\\frac{6s + 3s + s}{3} = 980\\\\6s + 3s + s = 980 \times 3\\\\10s = 2940\\\\s = 294[/tex]

Thus sarah has received £ 294

Final answer:

To solve for the amount Sarah receives, set up ratios based on the information provided and solve the resulting equation. The total amount divided among them is £980, which when divided by the total parts (10) gives Gavyn's share as £98. Sarah's share is three times Gavyn's share, resulting in £294.

Explanation:

The problem described is a classic division in ratio mathematics question where £980 is being divided among Caroline, Sarah, and Gavyn following certain rules.

According to the problem, Caroline receives twice the amount that Sarah receives and Sarah receives triple the amount that Gavyn receives.

We can set up the following ratios: C = 2S and S = 3G, where C stands for Caroline's amount, S for Sarah's, and G for Gavyn's. If we denote Gavyn's amount as G, then Sarah's amount is 3G and Caroline's is 2 × 3G which is 6G.

To find the value of G, we can write the equation G + 3G + 6G = £980 or 10G = £980. Solving for G gives us G = £98.

Therefore, Sarah, receiving three times as much as Gavyn, gets 3 × £98 = £294.

Which expression gives the area of the RED rectangle.




A

(A + B)(C + D)


B

(A + B)(C - D)


C

(A + B)(C + D) - C(A + B) - BD


D

(A + B)(C + D) - D(A + B) - BD

Answers

The expression which  gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD.  option C is correct.

The area of rectangle is obtained by multiplying the length and width.

The length of the rectangle is C+D

The width of the rectangle is A+B.

Now the complete area of rectangle :

Area = (A+B)(C+D)

Now to find area of rectangle which is red we have to subtract the red rectangle which are in blue:

The area of left side rectangle which is blue:

Area =C(A+B)

The area of rectangle below red rectangle:

Area =BD

So the area of red rectangle : (A+B)(C+D)-C(A+B)-BD.

Hence, option C is correct, the expression which  gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD.

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State if the triangles in each pair are similar. If so, State how you know they are similar and complete the similarity statement.

Answers

Answer:

Step-by-step explanation:

Looking at both triangles, angle P in triangle PQR = 38 degrees. Angle N in triangle LMN = 38 degrees. Both angles are equal.

Side PQ in triangle PQR = 16

Side MN in triangle LMN = 8

Therefore,

PQ/MN = 16/8 = 2

Side PR in triangle PQR = 14

Side LN in triangle LMN = 7

Therefore,

PR/LN = 14/7 = 2

Therefore, triangle PQR is similar to

triangle LMN because

1) the length of PQ is proportional to the length of MN.

2) the length of PR is proportional to the length of LN

3) angle P = angle N

4) Therefore, QR is also proportional to ML

Therefore,

PQ/MN = PR/LN = QR/ML = 2

write a verbal phrase to describe f > -4

Answers

I would say it as "the letter F is greater than negative four."

Find the m∠p.
pls and thanks <3

Answers

Answer:

36°

Step-by-step explanation:

Like your other question, the angles of the triangle must add up to 180. The tangent line is perpendicular to the center, so the angle must be 90°.

90° + 54° + 36° = 180°

Which association best describes the data in the table?

x y
3 2
3 3
4 4
5 5
5 6
6 7
8 8

A. no association

B. negative association

C. positive association

Answers

Answer:

'x'  and 'y' have a positive association between them.

Step-by-step explanation:

We can see in the given data that as 'x' remains  constant or increases by 1 unit, or two units in it's two consecutive values, in each of the cases 'y' increases by 1 unit.

Hence, 'x'  and 'y' have a positive association between them.

Answer:

C. positive association

Step-by-step explanation:

Positive and negative association describe a relation between variables about a scatter plot.

A positive association happens when one variable increases while the other one also increases.

A negative association happens when one variable decreases while the other variable increases.

It's important to know that the variable that always increases is the independent variable, it increases no matter what.

In this case, we could say that we have a positive association, because while x increases, y also increases.

You could notice that there's some repetitive number. That's normal because a scatter plot doesn't describe a perfect linear relation at the beginning, actually, it's a bit messy, however, from such data we construct a linear relationship which is called "linear regression".

Therefore, the right answer is C.

A couple needs $55,000 as a down payment for a home. If they invest the $40,000 they have at 4% compounded quarterly, how long will it take for the money to grow to $55,000? (Round your answer to the nearest whole number.)

Answers

Answer:

8 years

Step-by-step explanation:

Compound interest formula

[tex]A(t)= A_0(1+\frac{r}{n})^{nt}[/tex]

A(t) is the final amount 55000

A_0= 40000, r= 4% = 0.04, for quarterly n=4

[tex]55000=40000(1+\frac{0.04}{4})^{4t}[/tex]

divide both sides by 40000

[tex]1375=(1+\frac{0.04}{4})^{4t}[/tex]

[tex]1375=(1.01)^{4t}[/tex]

Take ln on both sides

[tex]ln(1375)=4tln(1.01)[/tex]

divide both sides by ln(1.01)

[tex]\frac{ln 1375}{ln 1.01}=4t[/tex]

Divide both sides by 4

t=8.00108

So it takes 8 years

Final answer:

The couple will need to invest their $40,000 at an interest rate of 4% compounded quarterly for about 7 years in order to reach their target of $55,000.

Explanation:

The subject of this question is compound interest. Compound interest is the interest computed on the initial principal as well as the accumulated interest from previous periods. Since the couple's money is being compounded quarterly, we will need to use this information in our calculations.

 

First, we must understand the compound interest formula which is:

A = P (1 + r/n)^(nt)

where,

A is the final amount of money after n years. P is the principal amount (initial amount of money). r is the annual interest rate in decimal form (so 4% would be 0.04). n is the number of times the interest is compounded per year. t is the time the money is invested for in years. In this case, we are trying to find 't' when A = $55,000, P = $40,000, r = 0.04 and n = 4 (since the interest is compounded quarterly). Doing the math, we get the answer as approximately 7 years.
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What is the measure of a°?

Answers

Answer:

The correct answer is C. 129.

Step-by-step explanation:

Let's recall that the Inscribed Quadrilateral Theorem states that a quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.

In the case of the inscribed quadrilateral ABCD in the graph attached, we have that:

m∠C= 51°, therefore its supplementary angle, ∠A, should be the difference between 180° and m∠C.

m∠C= 180° - m∠A

Replacing with the real values:

51 = 180 - m∠A

m∠A = 129°

The correct answer is C. 129.

The function f(x) = 6x + 8 represents the distance run by a cheetah in miles. The function g(x) = x − 2 represents the time the cheetah ran in hours. Solve (f/g)(3), and interpret the answer.
a. 26; the cheetah's rate in miles per hour
b. 26; the number of cheetahs
c. 1/26 ; the cheetah's rate in miles per hour
d. 1/26 ; the number of cheetahs

Answers

Option a. 26; the cheetah's rate in miles per hour is the correct answer.

Step-by-step explanation:

Given

[tex]f(x) = 6x+8\\g(x) = x-2[/tex]

As the function f is in miles and function g is is hours, and we are dividing the function f by function g so the new unit will be:

miles/hour = miles per hour

Now

[tex]\frac{f}{g}(x)= \frac{f(x)}{g(x)}\\\frac{f}{g}(x) = \frac{6x+8}{x-2}[/tex]

We have to find (f/g)(3) so putting 3 in place of x

[tex]\frac{f}{g}(x) = \frac{6(3)+8}{-2}\\= \frac{18+8}{1}\\= 26[/tex]

Hence,

Option a. 26; the cheetah's rate in miles per hour is the correct answer.

Keywords: Functions, function operations

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If you have a lowest score of 21 and a range of 47, your highest score will be:________

Answers

Answer: 68

Step-by-step explanation:

range=47 lowest score = 21 let highest score =x

Range=highest score - lowest score

47 = x - 21

x = 47 + 21

x = 68

Therefore the highest score is 68

The highest score is 68.

Range = Highest Score - Lowest Score

In this case,

the lowest score is 21 and

the range is 47.

So, we can rearrange the formula to solve for the highest score:

Highest Score = Range + Lowest Score

Substituting in the given values:

Highest Score = 47 + 21

Thus, the highest score will be 68.

An airplane flew 4 hours with a 25 mph tail wind. The return trip against the same wind took 5 hours. Find the speed of the airplane in still air. This similar to the current problem as you have to consider the 25 mph tailwind and headwind. Plane on outbound trip of 4 hours with 25 mph tailwind and return trip of 5 hours with 25 mph headwind Let r = the rate or speed of the airplane in still air. Let d = the distance

a. Write a system of equations for the airplane. One equation will be for the outbound trip with tailwind of 25 mph. The second equation will be for the return trip with headwind of 25 mph.
b. Solve the system of equations for the speed of the airplane in still air.

Answers

Answer:

r  = 225 Mil/h     speed of the airplane in still air

Step-by-step explanation:

Then:

d  is traveled distance   and r  the speed of the airplane in still air

so the first equation is for a 4 hours trip

as  d = v*t

d  =  4 *  ( r  + 25)    (1)          the speed of tail wind  (25 mil/h)

Second equation the trip back in 5 hours

d  =  5  * ( r  - 25 )    (2)

So we got a system of two equation and two unknown variables  d  and

r

We solve it by subtitution

from equation (1)     d  =  4r  + 100

plugging in equation 2

4r  + 100  = 5r -  125    ⇒   -r  =  -225     ⇒    r  = 225 Mil/h

And distance is :

d  =  4*r  +  100         ⇒ d  =  4 * ( 225)  +  100

d  =  900 +   100

d  = 1000 miles

Match the square root with its perfect square: 1 . √4 1 2 . √144 2 3 . √9 3 4 . √121 4 5 . √64 5 6 . √169 6 7 . √100 7 8 . √25 8 9 . √1 9 10 . √36 10 11 . √81 11 12 . √16 12 13 . √49 13

Answers

Answer:

√1 1

√4 2

√9 3

√16 4

√25 5

√36 6

√49 7

√64 8

√81 9

√100 10

√121 11

√144 12

√169 13

Step-by-step explanation:

1. = 49

2. = 169

3. = 81

4. = 100

5. = 441

6. = 36

I just finished the assignment, trust me.

The Davis family traveled 35 miles in 1/2 of an hour. If it is currently 2;00 pm and the family destination is 245 miles away at what time will they arrive there explain how you solved the problem

Answers

Answer:

5.30 PM

Step-by-step explanation:

time taken=(1/2)×(1/35)×245=7/2 hrs=3hrs 30 min.

so time =2+3.30=5.30 PM

Answer: 5:30pm

Step-by-step explanation:

Distance of family destination from the starting point = 245miles

Average speed = 35miles/1/2hour

Therefore 245 miles = 245/35 x 30mins.

7 x 30 = 210minutes.

Converted to hour by dividing by 60

210/60 = 3hours 30minutes.

Current local time at the point of commencement

Therefore, the arrival time at their destination = 14hours + 3hours 30mins.

= 17hours 30minutes. Convert to local time

17hours 30minutes = 12hours

= 5:30 pm in the evening

They have succeeded in spending 3hours 30minutes on the road. The final answer = 5:30pm.

Lucia flips a coin three times. What is the probability she gets (Head, tail, Head) in that order?

Answers

Answer: [tex]\dfrac{1}{8}[/tex]

Step-by-step explanation:

The total outcomes of tossing a coin = 2

The total number of possible outcomes of flipping coin three times =2 x 2 x 2 = 8

Favorable outcome= (Head, tail, Head) in order

i.e. Number of Favorable outcomes = 1

We know that , Probability = [tex]\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]

Therefore , The required probability [tex]=\dfrac{1}{8}[/tex]

Hence , the probability she gets (Head, tail, Head) in that order[tex]=\dfrac{1}{8}[/tex]

Nancy performs a full backup of her server every Sunday at 1 A.M. and differential backups on Mondays through Fridays at 1 A.M. Her server fails at 9 A.M. Wednesday. How many backups does Nancy need to restore?

Answers

Answer:

4

Step-by-step explanation:

Final answer:

To restore her server after a failure on Wednesday morning, Nancy would need to restore the full backup from Sunday, and then restore the differential backup from Tuesday.

Explanation:

In Nancy's case, she would need two backups to fully restore her server. These would be the full backup from Sunday and the differential backup from Tuesday. Here's why:

A full backup involves copying all of the data in a system. It's the most comprehensive type of backup but also requires the most storage space and time. A differential backup, on the other hand, only backs up the data that has changed or been added since the last full backup.

Because Nancy performs full backups every Sunday, the full backup will have all the data up until Sunday at 1 A.M. The differential backup from Tuesday will contain all the changes that occurred on Monday and Tuesday until 1 A.M.

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Let A be a set with a partial order R. For each a∈A, let Sa= {x∈A: xRa}. Let F={Sa: a∈A}. Then F is a subset of P(A) and thus may be partially ordered by ⊆, inclusion.

a) Show that if aRb, then Sa ⊆ Sb.
b) Show that if Sa ⊆ Sa, then aRb.
c) Show that if B⊆A, and x is the least upper bound for B, then Sx is the least upper bound for {Sb:b∈B}

Answers

Answer:

See proofs below

Step-by-step explanation:

a) Suppose that aRb. Let y∈Sa , then y∈A and yRa. We have that yRa and aRb. Since R is a partial order, R is a transitive relation, therefore yRa and aRb imply that yRb. Now, y∈A and yRb, thus y∈Sb. This reasoning applies for all y∈Sa that is, for all y∈Sa, y∈Sb, threrefore Sa⊆Sb.

b) Suppose that Sa⊆Sb. Since R is a partial order, R is a reflexive relation then aRa. Thus, a∈Sa. The inclusion Sa⊆Sb implies that a∈Sb, then aRb.

c) Denote this set by S={Sb:b∈B}. We will prove that supS=Sx with x=supB.

First, Sx is an upper bound of S: let Sb∈S. Then b∈B and since x=supB, x is an upper bound of B, then bRx. Then b∈Sx. Now, for all y∈Sb, yRb and bRx, then by transitivity yRx, thus y∈Sx. Therefore Sb⊆Sx for all Sb∈S, which means that Sx is an upper bound of S (remember that the order between sets is inclusion).

Now, let's prove that Sx is the least upper bound of S. Let Sc⊆A be another upper bound of S (in the set F). We will prove that Sx⊆T.

Because Sc is an upper bound, Sb⊆Sc for all b∈B. Thus, if y∈Sb for some b, then y∈Sc. That is, if yRb then yRc. In particular, bRb then bRc for all b∈B. Thus c is a upper bound of B. Byt x=supB, then xRc. Now, for all z∈Sx, zRx and xRc, which again, by transitivity, implies that z∈Sc. Therefore Sx⊆Sc and Sx=sup S.

Final answer:

In a partial order set, if aRb, then Sa ⊆ Sb. If Sa ⊆ Sb, then aRb. If B ⊆ A and x is the least upper bound for B, then Sx is the least upper bound for {Sb : b ∈ B}.

Explanation:

a) To show that if aRb, then Sa ⊆ Sb, we need to prove that if x ∈ Sa, then x ∈ Sb. Since x ∈ Sa, that means xRa holds. And since aRb, we can conclude that xRb holds as well. Therefore, x ∈ Sb, which implies that Sa ⊆ Sb.

b) To show that if Sa ⊆ Sb, then aRb, we need to prove that if aRb does not hold, then Sa ⊆ Sb does not hold. If aRb does not hold, it means that b is not in Sa. However, if Sa ⊆ Sb, it implies that every element in Sa is also in Sb. Hence, we arrive at a contradiction, which proves that if Sa ⊆ Sb, then aRb.

c) To show that if B ⊆ A and x is the least upper bound for B, then Sx is the least upper bound for {Sb : b ∈ B}, we need to prove that Sx is an upper bound for {Sb : b ∈ B} and that it is the least upper bound. Since x is the least upper bound for B, this means that every element of B is contained in Sx. Therefore, Sx is an upper bound for {Sb : b ∈ B}. Additionally, if there exists an upper bound U for {Sb : b ∈ B}, then every element of {Sb : b ∈ B} must be contained in U. Since Sb is a subset of Sa for every b ∈ B, it follows that every element of Sa is contained in U. Therefore, Sa ⊆ U for every a ∈ A. In particular, Sx ⊆ U, which proves that Sx is the least upper bound for {Sb : b ∈ B}.

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Working alone at its constant rate, pump X pumped out \small \frac{1}{3} of the water in a pool in 4 hours. Then pump Y started working and the two pumps, working simultaneously at their respective constant rates, pumped out the rest of the water in 6 hours. How many hours would it have taken pump Y, working alone at its constant rate, to pump out all of the water that was pumped out of the pool?

Answers

Answer:

P(y) take 36 h to do the job alone

Step-by-step explanation:

P(x) quantity of water pump by Pump X     and

P(y)  quantity of water pump by Pump Y

Then if  P(x)  pumped  1/3  of the water in a pool in 4 hours

Then in 1 hour P(x) will pump

    1/3    ⇒  4 h

    ? x    ⇒   1 h       x  =  1/3/4     ⇒   x  = 1/12

Then in 1 hour   P(x)  will pump  1/12 of the water of the pool

Now both pumps   P(x)  and P(y)   finished 2/3 of the water in the pool (left after the P(x) worked alone ) in 6 hours. Then

P(x)  +  P(y)  in    6 h    ⇒  2/3

                     in    1 h   ⇒    x ??      x  =  (2/3)/6     x  =  2/18    x  = 1/9

Then   P(x)  +  P(y)  pump   1/9 of the water of the pool in 1 h. We find out how long will take the two pumps to empty the pool

water in a pool is  9/9  ( the unit) then

  1  h    ⇒   1/9

   x ??  ⇒   9/9        x  = ( 9/9)/( 1/9)     ⇒   x  =  9 h

The two pumps would take 9 hours working together from the beggining

And in 1 hour  of work, both  pump 1/9 of the water, and P(x) pump 1/12 in 1 hour

Then in 1 hour P(y)

P(y)   =  1/9  -  1/12    ⇒    P(y)   = 3/108         P(y)   = 1/36

And to pump all the water  (36/36) P(y) will take

  1 h     1/36

  x ??   36/36         x  =  (36/36)/1/36

 x =  36 h

P(y) take 36 h to do the job alone

Flip a coin 5 times. What is the probability that heads never occurs twice in a row?

Answers

Answer:

[tex]\frac{15}{32}[/tex]

Step-by-step explanation:

Given that a coin is flipped 5 times. Let us assume it is a fair coin with probability for head or tail equally likely and hence 0.50

Sample space would have [tex]2^5 = 32[/tex] possibilities

For two heads never occurring in a row favourable outcomes are

HTTTT,HTTHT,HTTTH,  HTHTH,  HTHTT, THTHT, THTTT, THTTH, TTHTH, TTHTT, TTTHT, TTTTH, TTTTT, TTTTH

Hence probability =

Favorable outcomes/Total outcomes=

[tex]\frac{15}{32}[/tex]

if a negative number has to be added to another negative number, does it stay negative?

Answers

Yes because the number say on one side

yes because the numbers stay on one side of the number line

A vegetable garden and its surrounding that are shaped like a square that together are 11 feet wide. The path is 2 feet wide. Find the total area of the vegetable garden and path. ​

Answers

Area of path and garden = 11 x 11 = 121 square feet.

Area of garden only = 7 x 7 = 49 square feet.

Area of just the path = 121 - 49 = 72 square feet.

Ann increased the quantities of all the ingredients in a recipe by 60\%60%60, percent. She used 808080 grams (\text{g})(g)left parenthesis, start text, g, end text, right parenthesis of cheese. How much cheese did the recipe require?

Answers

Answer:

50 grams

Step-by-step explanation:

Let the amount of cheese required by the recipe be "x"

Ann increased 60% from original amount and then used up 80 grams. Thus:

Original, increased by 60%, became 80

This translated to algebraic equation would be:

x + 0.6x = 80

Note:  60% = 60/100 = 0.6

So we can solve the above equation for "x" and get our answer. Shown below:

[tex]x + 0.6x = 80\\1.6x=80\\x=\frac{80}{1.6}\\x=50[/tex]

Hence,

the recipe required 50 grams of cheese

Jasmine weigh 150ib he is loading a freight elevator with identical 72-pound boxes. The elevator can carry no more than 2000ib. If Jasmine rides with the boxes,how many boxes can be loaded on the elevator?

Answers

Answer:

Jasmine can load maximum of 25 boxes with herself on the elevator.

Step-by-step explanation:

Given:

Weight of Jasmine = 150 lb

Weight of each boxes  = 72 lb

Load elevator can carry = 2000 lb

we need to find the number of boxes that can be loaded

Let number of boxes be 'x'

Now we know that Maximum load the elevator can carry is 2000 lb.

So We can say Weight of jasmine plus Number of boxes multiplied by Weight of each boxes should be less than or equal to Load elevator can carry.

Framing in equation form we get;

[tex]150+72x\leq 2000[/tex]

Solving the equation we get:

We will first Subtract 150 on both side;

[tex]150+72x-150\leq 2000-150\\\\72x\leq 1850[/tex]

Now Dividing both side by 72 by using Division property we get;

[tex]\frac{72x}{72}\leq \frac{1850}{72}\\\\x\leq 25.69[/tex]

Hence Jasmine can load maximum of 25 boxes with herself on the elevator.

Here are the records of two different sequences (A,B ) of a coin tossed eight times. A: T H H H H H H H B: H T T H T H H T If you know for sure that the coin is fair, are these two sequences equally probable outcomes or, if they are not, which sequences is more probable than the other?"

Answers

Answer: Sequence B is more probable than A.

Step-by-step explanation:

This two sequences are not equally probable. Sequence B is more probable than A due to the equal chances of getting head (H) and a tail (T). The probability of getting a head is equal to the probability of getting a tail which is 4/8 i.e 0.5

The sequence A is less probable because the head(H) occur more than tail (T). The probability of head occurring is almost a sure event i.e 1 which is not feasible.

Three pounds of dried cherries cost $15.90, 5 pounds of dried cherries cost $26.50, and 9 pounds of dried cherries cost $47.70. Which equation gives the total cost y of x pounds of dried cherries?

Answers

Final answer:

The equation that gives the total cost of x pounds of dried cherries is y = (10.60/2)x.

Explanation:

Let's create a table to represent the given information:

Pounds of Dried Cherries (x)Total Cost (y)315.90526.50947.70

To find an equation that gives the total cost of x pounds of dried cherries, we need to find a pattern in the data. From the table, we can see that as the pounds of dried cherries increase, the total cost also increases. This suggests a linear relationship between pounds and cost.

Now, let's analyze the changes in cost for each additional pound of dried cherries:

From 3 to 5 pounds: The cost increased by $26.50 - $15.90 = $10.60.From 5 to 9 pounds: The cost increased by $47.70 - $26.50 = $21.20.

Based on these changes, the cost increased by $10.60 for every 2 additional pounds of dried cherries. Therefore, the equation that gives the total cost y of x pounds of dried cherries is:

y = (10.60/2)x

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