Let r and s be the usual generators for the dihedral group of order 8...
Part a)
list the elements of D8 as 1,r,r^2,r^3,s,sr,sr^2,sr^3 and label these with the integers
1,2,...,8 respectively. Exhibit the image of each element of D8 under the left regular representation of D8 into S8....
Answers
Each element of D8 is mapped to a permutation of {1, 2, 3, 4, 5, 6, 7, 8} by considering its action on the labeled generators 1, 2, ..., 8 under composition.
The elements of D8 are 1, r,\(r^2, r^3, s, sr, sr^2, sr^3,\) which we can label with the integers 1, 2, 3, 4, 5, 6, 7, 8 respectively. The left regular representation of D8 into S8 is given by:
1 → (1)(2)(3)(4)(5)(6)(7)(8)
r → (1 2 3 4)(5 6 7 8)
\(r^2\) → (1 3)(2 4)(5 7)(6 8)
\(r^3\)→ (1 4 3 2)(5 8 7 6)
s → (1 2)(3 4)(5 6)(7 8)
sr → (1 8 3 6)(2 7 4 5)
\(sr^2\)→ (1 4)(2 3)(5 8)(6 7)
\(sr^3\) → (1 6 3 8)(2 5 4 7)
Here, each element of D8 is mapped to a permutation of {1, 2, 3, 4, 5, 6, 7, 8} by considering its action on the labeled generators 1, 2, ..., 8 under composition.
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We have exhibited the image of each element of D8 under the left regular representation into S8, represented as a product of disjoint cycles.
The left regular representation of D8 is a homomorphism from D8 to the symmetric group S8, where each element of D8 is represented as a permutation of the set {1, 2, ..., 8}.
Using the generators r and s, we can construct the elements of D8 as follows:
\(r^0\) = 1
\(r^1\)= r
\(r^2\) = rr
\(r^3\) = rrr
\(s^0\)= 1
\(s^1\) = s
\(s^2\) = 1
\(s^3\) = ss
Now, let's apply the left regular representation to each element of D8:
1 → (1 2 3 4 5 6 7 8)
r → (1 2 3 4 5 6 7 8)(1 2)
\(r^2\) → (1 2 3 4 5 6 7 8)(1 3)(2 4)
\(r^3\)→ (1 2 3 4 5 6 7 8)(1 4 3 2)
s → (1 8)(2 7)(3 6)(4 5)
sr → (1 7 3 5)(2 8 4 6)
\(sr^2\) → (1 6 3 2)(4 7 5 8)
\(sr^3\) → (1 5)(2 6)(3 7)(4 8)
Note that we can represent each permutation as a product of disjoint cycles, where each cycle corresponds to an orbit under the action of the permutation. For example, the permutation corresponding to r is a product of two cycles: (1 2) and (3 4 5 6 7 8). This means that r fixes the elements 1 and 2, and permutes the elements 3, 4, 5, 6, 7, and 8 cyclically. Similarly, the permutation corresponding to s is a product of four cycles: (1 8), (2 7), (3 6), and (4 5), which means that s fixes the pairs (1, 8), (2, 7), (3, 6), and (4, 5), and transposes each pair.
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Related Questions
What is the coefficient in the equation 8 = 4 + 2g?
9
2
4
8
Answers
Step-by-step explanation:
8=4+2g
8-4=2g
g=4/2
g=2
Answer:
2 is the coefficient
Step-by-step explanation:
2 because it's before the variable
Let me know if this helped
Pleaseee help IM stuck
Answers
Answer:
The first one
Step-by-step explanation:
I think that for you to convert miles to feet you would need to cancel out miles which means miles goes in the bottom. I'm not sure though
Calculate the area of the circle.
11 mm
Give your answer correct to 1 decimal place.
Answers
Answer: 95.0
Step by step explanation:
Answer:
95.07142857
Step-by-step explanation:
Given: diameter (d) = 11 mm
radius (r) =
\( \frac{d}{2} \)
= 5.5 mm
Now,
Area of circle=
\(\pi \: r {}^{2} \)
=22/7 * (5.5)^2
=95.07142857
What is the simplified form of the square root of 64x16? 8x4 8x8 32x4 32x8
Answers
Answer:
8
Step-by-step explanation:
Answer:
8x⁴
Step-by-step explanation:
8x8 = 64
⁴x⁴ = ¹⁶
A line passes through the points (1, -6) and (4, 3).
10
8-
What is the v-intercent of this line?
Answers
Answer:
The y-intercept is -9.
Step-by-step explanation:
To find the y-intercept, you first need to find the slope of the two points. (1,-6) and (4,3). Using rise over run the slope you should get is 3. To find the y-intercept we can use the y = mx + b.
Sorry, if I didn't understand the question, I hope this helps.
find the inversef(n)=-3/ n+1 +1
Answers
In order to find the inverse of the function f(n), we just need to switch f(n) by n, and switch n by f^-1(n) in the function. Then, we isolate f^-1(n). So we have that:
\(\begin{gathered} f(n)=-\frac{3}{n+1}+1 \\ n=-\frac{3}{f^{-1}(n)+1}+1 \\ -\frac{3}{f^{-1}(n)+1}=n-1 \\ \frac{3}{f^{-1}(n)+1}=1-n \\ 3=(f^{-1}(n)+1)(1-n) \\ f^{-1}(n)+1=\frac{3}{1-n} \\ f^{-1}(n)=\frac{3}{1-n}-1 \end{gathered}\)So the inverse of f(n) is 3/(1-n) - 1
Suppose that the random variables X and Y are independent and you know their distributions.
Which of the following explains why knowing the value of X tells you nothing about the value of Y?
A.
X and Y might be independent.
B.
The mean of X might be different from the mean of Y.
C.
The variance of X might be different from the variance of Y.
D.
All of the above.
Answers
A. X and Y might be independent. Knowing the value of one independent variable tells us nothing about the value of the other independent variable. The mean and variance of X and Y being different does not necessarily mean that knowing the value of one tells us nothing about the other.
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Betty and Bill just won $10,000 in the Pennsylvania state lottery. They decide to spend $3,000 now and put the remaining $7,000 in an investment earning 8 percent compounded annually. If they use the money in that investment for a vacation in five years, approximately how much will they have available to spend on that vacation?
Answers
Answer:
$10285.29
Step-by-step explanation:
The idea is to solve for the final amount when $7000 is invested and compounded
given data
principal= $7000
rate= 8%= 0.08
time =5years
We know that the expression for the final amount is
\(A = P(1 + {r)^{t}\)
A = 7000(1 + {0.08)^5
A=7000(1.08)^5
A=7000(.469328)
A= 10285.29
They have available to spend on that vacation $10285.29
What is the value of Arctan1? a.π b.4π c.π/4 d.4
Answers
Answer:
pi/4
Step-by-step explanation:
Solve for .
- 9x + 5 < 17
OR
13r + 25 < -1
Answers
x > −4/3
OR
For 13r + 25 <-1 it’s
R<-2
I NEED AN ANSWER ASAP PLEASE
Answers
Step-by-step explanation:
the correct answer is option a 6a-7
can someone help me pls
Answers
Answer:
i think you add them not sure...
Step-by-step explanation:
40+44+44+??
ok?
Answer:
x=88°
x-44°=88°-44°=44°
Step-by-step explanation:
angle A+ angle B+angle C=180°
48°+x-44°+x°=180°
=48°-44°+x°+x°=180°
=4°+2x°=180°
=4°-4°+2x°=180°-4°
=2x°=176°
=2x°/2=176°/2
=x°=88°
x°-44°=88°-44°
=x°-44°=44°
evaluate the function f(x)=5x+2 for f(-4)
Answers
Answer:
The answer is -18
Step-by-step explanation:
You plug in -4 and solve
f(-4)= 5(-4) +2
Use the given information to prove that DE=FD
Answers
The condition for the figure to be similar is that; DE = FD
What are similar figures?Similar figure are zoomed in or zoomed out (or just no zoom) version of each other. They are scaled version of each other, and by scale, we mean that each of their dimension(like height, width etc linear quantities) are constant multiple of their similar figure.
In the problem, the ratio should be CD/DE = GD/DF
We can also write the ratios of the corresponding sides as:
CD/DE = GD/DF
Therefore, CE and GF are similar
Thus, the condition for the figure to be similar is that;
DE = FD
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A school is arranging a field trip to the zoo. The school spends 673.86 dollars on passes for 31 students and 2 teachers. The school also spends 267.53 dollars on lunch for just the students. How much money was spent on a pass and lunch for each student?
Answers
Answer:
$21.73 for a pass, and $8.63 for a lunch
Step-by-step explanation:
673.81/31= 21.73
267.53/31=8.63
Hope this helps
The function F(x) =
3
(x – 3)(x + 3)
is never equal to zero.
O A. True
O B. False
SUBM
Answers
Answer: False
Explanation:
f(x) = 0
3(x-3)(x+3) = 0
x-3 = 0 or x+3 = 0
x = 3 or x = -3
This shows that if x is equal to either -3 or 3, then f(x) is 0.
andace had scores of 96, 84, 95, and 82 on her first four exams of the semester. what score must she obtain on the fifth exam to have an average of 90 or better for the five exams
Answers
Andace needs to score at least 98 on her fifth exam to have an average of 90 or better for the five exams.
To find out what score Andace needs to get on the fifth exam, we can use the formula for the average (also known as the mean):
Average = (sum of all scores) / (number of scores)
We know that Andace needs an average of 90 or better for the five exams. Therefore, the sum of all five scores must be at least 450 (90 multiplied by 5).
The sum of Andace's first four scores is:
96 + 84 + 95 + 82 = 357
To get an average of 90 or better for the five exams, Andace needs to get a total of:
450 - 357 = 93
on her fifth exam.
Since she has already taken four exams and the maximum score she can get on her fifth exam is 100, we can subtract her first four scores from 93 to find out what she needs to score on her fifth exam:
93 - 96 - 84 - 95 - 82 = -164
Since it's not possible for Andace to score a negative number on her fifth exam, we can conclude that she needs to score at least 98 (93 - 5) to have an average of 90 or better for the five exams.
In conclusion, Andace needs to score at least 98 on her fifth exam to have an average of 90 or better for all five exams.
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Problem 6 (Instrumental Variables, Theory - 8 points) a. Is the following reasoning or statement correct? Explain. Suppose that X is an endogenous variable for which we need an instrument. An elementary example of an instrumental variable that is relevant but not exogenous is the variable Xitself. b. Consider a two-stage least squares regression without any exogenous variables ( W 's). In the first stage X is instrumented by variables Z1 ,Z2, Z3, and Z4. The second stage result is: Y^=4.8−0.5 X^
(1.7) (0.11) The first stage was run without a constant. The first stage overall F-statistic is 16.7 (this is the F. statistic that tests whether all first-stage parameters are zero). The overidentifying 1 -test is 13.45. Now suppose that there were exogenous variables (W's) in the model. The first stage is still run without a constant. Do you use the F-statistic as described above (testing whether all of the parameters in the first stage are zero) to decide whether the instruments are relevant? Or do you use a different F-statistic? (Hint: the question is long, but the answer is two lines! Consider: what does it mean that the instruments are relevant? Does the definition of relevance, and in particular the rule of thumb that checks it, include the exogenous variables (W's)?)
Answers
a. X can be an instrument for itself if relevant.
b. Exogenous variables don't affect instrument relevance.
a. The reasoning or statement is correct. X can serve as an instrumental variable for itself in certain cases. This occurs when X is endogenous (correlated with the error term in the regression equation) and there is a valid reason to believe that X affects the outcome variable but is not directly affected by other factors.
b. In the presence of exogenous variables (W's) in the model, the F-statistic described above, which tests whether all first-stage parameters are zero, is not used to determine the relevance of instruments. The concept of relevance of instruments is concerned with whether the instruments are correlated with the endogenous variable (X) in the first stage. Exogenous variables (W's) are not part of the relevance assessment because they are assumed to be unrelated to the endogenous variable. Therefore, the F-statistic used to test relevance would only consider the instruments (Z1, Z2, Z3, Z4) and their correlation with X, disregarding the presence of exogenous variables.
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1. A city planner is mapping out some new features for a triangular park. She sketches the park on grid paper. The coordinates of the vertices of the park are:
(3, -2), (3, 4), and (−3, 4).
The city planner wants to put a tree at the orthocenter of the park. What are the coordinates of the tree?
2. The city planner wants to put a flower bed at the centroid of the park. What are the coordinates of the tree?
3. picture below
Answers
Answer:
Download Socratic it has all the information you need
Whit is the general form of the equation of the given circle with center A
Answers
The general form of the equation of the given circle with center A is (x − h)² + (y − k)² = a².
What is a circle?A circle is a two-dimensional object made up of points that are spaced out from a given point (center) on the plane by a fixed or constant distance (radius).
Given:
The circle with center A
The general form of the equation of a circle is: x2 + y2 + 2gx + 2fy + c = 0.
The equation of the circle is given by (x−h)² + (y−k)² = a²
Here, h and k are the x and y coordinates of the circle and a is the radius of the circle.
Thus, the standard equation of a circle is (x−h)² + (y−k)² = a².
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Simplify 2s + 4 + (-6s) + 10
-4s + 14
4s + 14
10s
18s
Answers
Answer:
The answer is -4s + 14
Step-by-step explanation:
First what you want to do is add the -6s to 2s they have the same variables then add 4 to 10 which equals 14. I hope that helped! :)
What value of m would make parallelogram wxyz a square.
Answers
To make parallelogram WXYZ a square, the value of m should be 90 degrees.
A square is a special type of parallelogram in which all angles are right angles (90 degrees). In a parallelogram, opposite angles are congruent, which means they have the same measure. To transform the parallelogram WXYZ into a square, we need to make sure that all angles of the parallelogram are right angles.
By setting m = 90 degrees, we ensure that all four angles of parallelogram WXYZ are right angles. This condition will make the parallelogram into a square. A square has congruent sides and diagonals that bisect each other at right angles.
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Iarrecorded the grade-level and instrument of everyone in the middle school Schoolof Rock below.Seventh Grade StudentsEighth Grade StudentsInstrument # of StudentsInstrument # of Students1279GuitarBassDrumsKeyboard14GuitarBassDrumsKeyboard4355Based on these results, express the probability that a seventh grader chosen atrandom will play the drums as a decimal to the nearest hundredth.
Answers
Let's take a look at the seventh graders table.
Seventh grade has a total of
\(12+9+4+5=30\)30 students.
Out of those 30 students, 4 play the drums. Therefore, the probability that a seventh grader chosen at random will play the drums is:
\(\frac{4}{30}\rightarrow\frac{2}{15}\rightarrow0.13\)1. Fernando is using a garden hose to fill his backyard pool at a rate of 10 gallons per minute. The pool already contains 9000 gallons of water. The capacity of the pool is 12,000 gallons. a. Define the independent and dependent variables. b. Define the unit rate of change. 2. Use double number lines to solve each equation. a. 2(x + 1) = 10 b. −(x − 5) = 0 3. Solve for each unknown. a. 1.6 = x 2.8 7 b. 2 = 0.4 d4 5
Answers
The independent variable is the time in minutes.
The dependent variable is the capacity (volume) of the pool.
The unit rate of change is equal to 10.
The solution to each equation is shown on the number line below.
The solutions to each equations 1.6/2.8 = x/7 and 2/d = 0.4/5 are 4 and 25 respectively.
What is an independent variable?In Mathematics, an independent variable can be defined as the variable that is being manipulated by an experimenter, scientist, or a researcher. This ultimately implies that, an independent variable is always considered as the cause in an experiment and it is represented by the x-axis value on a graph.
In this context, we can reasonably infer and logically deduce that time measured in minutes represent the independent variable while the capacity (volume) of the pool represent the dependent variable.
Next, we would solve the equation algebraically;
1.6/2.8 = x/7
2.8x = 11.2
x = 11.2/2.8
x = 4.
2/d = 0.4/5
0.4d = 10
d = 10/0.4
d = 25.
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Ezra has $20 to spend on a taxi ride. The taxi company charges a flat rate of $3, plus $1.00 per mile. What is the maximum amount of miles Ezra can afford to travel in this taxi?
Answers
Answer:
17 miles
Step-by-step explanation:
20 - 3 = 17
(i may have misunderstood the question, if its $4 every mile then its 5 miles maximum)
a diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. also, if a person does not have the disease, the test will report that he or she does not have it with probability 0.9. only 3% of the population has the disease in question. if a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? (round your answer to four decimal places.) are you surprised by the answer? would you call this diagnostic test reliable? this answer has not been graded yet.
Answers
The conditional probability that a person actually has the disease given a positive test result is approximately 0.0826 (rounded to four decimal places). This probability suggests that the test is not highly reliable.
To determine the conditional probability, we can use Bayes' theorem. Let's denote D as the event that a person has the disease and T as the event that the test result is positive. We are interested in finding P(D|T), the probability that a person has the disease given a positive test result.
According to the problem statement, P(D) = 0.03, which represents the probability that a person chosen at random from the population has the disease. The test has a sensitivity of 0.9, meaning that P(T|D) = 0.9, indicating that the test correctly detects the disease in 90% of individuals who actually have it.
Additionally, the test has a specificity of 0.9, implying that P(T|D') = 0.9, where D' represents the event that a person does not have the disease.
We can now apply Bayes' theorem:
P(D|T) = (P(D) * P(T|D)) / [P(D) * P(T|D) + P(D') * P(T|D')]
Substituting the known values:
P(D|T) = (0.03 * 0.9) / [(0.03 * 0.9) + (0.97 * 0.1)]
Calculating this expression yields P(D|T) ≈ 0.0826, or approximately 8.26%.
Given this result, it can be observed that the conditional probability of having the disease, given a positive test result, is relatively low. This suggests that the test is not highly reliable.
Although the test has a high accuracy in correctly identifying individuals without the disease, it has a significant false-positive rate when it comes to individuals who do have the disease. Therefore, it is important to consider other factors and confirmatory tests before reaching a conclusive diagnosis.
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Does anybody know how i can choose a subject in this app
Answers
Answer:
You can click *BRAINLY* the symbol, which takes you to brainly home. Click the dropdown next to the little carrot and choose your desired subject.
Hope this Helps!
In two years, the value of a car changed from $14,000 to $11,200.What is the percent of change
Answers
Answer:
Percentage change = (14000-11200) / 11200 * 100
= 280000 / 11200
= 25%
So, your final answer is 25%
Hope this helps!
Step-by-step explanation:
3) help with easy math equation
Answers
Answer:
add 2
Step-by-step explanation:
Hi there!
r=4
1.Add the total and the other term:
\(14+2=16\)
2.Divide the two terms:
\(16*4=4\)
Therefore, r=4.
b. Use ratio language to write two different sentences that compare the amount
of ribbon and the number of bouquets.
Answers
The ratio of ribbon to bouquets is 2:5, meaning for every 2 meters of ribbon, 5 bouquets can be made. For every 7 bouquets, 3 meters of ribbon are needed, meaning the ratio is 3:7.
What is ratio?A ratio is a comparison of two numbers or quantities that are related to each other in some way. It is the relationship between the sizes of two or more things, expressed as the quotient of one divided by the other.
Ratios can be used to compare different aspects of a set of data, such as the ratio of boys to girls in a classroom, or the ratio of red cars to blue cars on a street. They are also used in many real-world applications, such as finance, engineering, and science.
Ribbon to bouquets ratio is 2:5, which means that for every 2 units of ribbon, there are 5 bouquets.
With a ribbon to bouquet ratio of 1:3, the number of bouquets is three times the amount of ribbon.
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