The power series solutions for the Hermite equation of 2 are zero for the first four terms of the given equation.
Equation = y''-2xy'+4y=0
The solutions can be expressed as power series using the Hermite equation of degree 2 can be calculated as:
y = ∑(n=0 to ∞) [tex]a_n x^{n}[/tex]
where [tex]a_n[/tex] is the coefficient of the nth term and x is the variable.
Differentiating y with regard to x,
y = ∑(n=0 to ∞) [tex]a_n x^{n-1}[/tex]
Double integrating the y with respect to x:
y'' = ∑(n=0 to ∞) [tex]a_nn(n-1)x^{n-2}[/tex]
Substituting the above equation in the Hermite equation
∑(n=0 to ∞) [tex]a_nn(n-1)x^{n-2}[/tex] - 2x∑(n=0 to ∞) [tex]a_n x^{n-1}[/tex] + 4∑(n=0 to ∞) [tex]a_n x^{n}[/tex] = 0
∑(n=0 to ∞)[tex][a_n(n(n-1) - 2n + 4)] x^{n}[/tex] = 0
Taking the coefficients of each term as zero:
[tex]a_n[/tex](n(n-1) - 2n + 4) = 0
The first four non-zero terms:
If n = 0,
[tex]a_o[/tex](0(0-1) - 2(0) + 4) = 0
[tex]a_o[/tex](4) = 0
[tex]a_o[/tex] = 0
If n = 1,
[tex]a_1[/tex](1(1-1) - 2(1) + 4) = 0
[tex]a_1[/tex](2) = 0
[tex]a_1[/tex] = 0
If n= 2,
[tex]a_2[/tex](2(2-1) - 2(2) + 4) = 0
[tex]a_2[/tex](2) = 0
[tex]a_2[/tex]= 0
If n = 3,
[tex]a_3[/tex] (3(3-1) - 2(3) + 4) = 0
[tex]a_3[/tex] (2) = 0
[tex]a_3[/tex] = 0
Therefore we can conclude that the power series solutions for the Hermite equation of 2 are zero.
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Compute r''(t) and r'''(t) for the following function. r(t) = (9t² +6,t + 5,6) Find r'(t). r(t) = 0.00
The derivatives of the vector function are r'(t) = (18 · t, 1, 0), r''(t) = (18, 0, 0) and r'''(t) = (0, 0, 0).
How to determine the first three derivatives in a vector functionIn this question we find the definition of a vector function in terms of time, whose first, second and third derivatives must be found. This can be done by using derivative rules several times. First, define the vector function:
r(t) = (9 · t² + 6, t + 5, 6)
Second, find the first derivative:
r'(t) = (18 · t, 1, 0)
Third, find the second derivative:
r''(t) = (18, 0, 0)
Fourth, find the third derivative:
r'''(t) = (0, 0, 0)
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Saul decides to use the IQR to measure the spread of the data. Saul calculates the IQR of the data set to be . Saul asks Jasmine to check his work. Saul copies the data in numerical order from least to greatest for Jasmine.
The question is incomplete. The compete question is :
Saul decides to use the IQR to measure the spread of data. Saul calculates the IQR of the data set to be 27. Saul asks Jasmine to check his work. Saul copies the data in numerical order from least to greatest for Jasmine. 25, 30, 50, 50, 50, 50, 56, n, 250. What is the value of n that will make the IQR of the data set equal to 27?
Solution :
The inter quartile range is being measured as : 3rd quartile - 1st quartile
The 1st quartile = 25% mark
The 3rd quartile = 75% mark
The data set as given in the question is :
25, 30, 50, 50, 50, 50, 56, n, 250
Therefore, the total number of the sample = 9
And the median is = 50
The median separates the given data sets into two equal halves-- that is the upper half and the lower half.
Let the middle of a lower half is the 1st quartile be [tex]$Q_1$[/tex]
And the middle of a 2nd half is the 3rd quartile be [tex]$Q_3$[/tex]
∴ [tex]$Q_1=\frac{30+50}{2}$[/tex]
[tex]$=40$[/tex]
[tex]$Q_3=\frac{56+n}{2}$[/tex]
Now for the IQR of the data set to be equal to [tex]$27$[/tex],
[tex]$\frac{56+n}{2}-40=27$[/tex]
[tex]$\frac{56+n}{2}=27+40$[/tex]
[tex]$\frac{56+n}{2}=67$[/tex]
[tex]$56 +n=134$[/tex]
[tex]$n=134-56$[/tex]
[tex]$n=78$[/tex]
Please help me, GodBless.
Answer:
Rate of change is 1
Step-by-step explanation:
Well rate of change in a function is basically slope
theres 1 rise and.1 run
1/1=1
So rate of change is 1
Help pls I’m on a time limit. Please and thank you <33
Answer: the answer is 3/10
Step-by-step explanation: because its 3/10
Answer:
es
3/ 10
Step-by-st
You invest $32,000 in an account that earns 5.75% interest compounded yearly. What is
the total amount of money you will have in the account after 20 years if you never
deposit any additional funds?
Answer:
$97,894.33.
Step-by-step explanation
The G.M and H.M between two number are respectively 9 & 5.4. Find the numbers.
Answer:
The numbers are 3 and 27.
Step-by-step explanation:
The explanation is attached below.
What is the probability of 4 consecutive 2-sided coin tosses all coming up heads?
The probability of getting four consecutive heads in four 2-sided coin tosses is 1/16 or 0.0625 or 6.25%.
The Probability is defined as a measure of the likelihood or chance that an event will occur, expressed as a value between 0 and 1, where 0 represents impossibility and 1 represents certainty.
To calculate the probability of getting four consecutive heads in a row, we multiply the probabilities of each individual toss.
The Probability of heads (H) is = 1/2,
So, The Probability of four consecutive heads (HHHH) is :
= (1/2) × (1/2) × (1/2) × (1/2) = (1/2)⁴ = 1/16,
Therefore, the required probability is 1/16 or 6.25%.
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Simplify.
+6 - 4 =
please help ..........
Answer:
2
Step-by-step explanation:
it's literally 6-4 = 2
Answer: 2
Step-by-step explanation: +6 is simply 6 and 6-4 is 2.
What’s the website in this image called?
A 5 meter ladder is leaning against a house when its base starts to slide away. By the time the base is 3 meter from the house, the base is moving at the rate of 2 m/sec. (a) How fast is the top of the ladder sliding down the wall then? (b) At what rate is the angle between the ladder and the ground changing then?
(a) The top of the ladder is sliding down the wall at a rate of 3/2 m/sec.
(b) The angle between the ladder and the ground is changing at a rate of 1/2 rad/sec.
To solve this problem, we can use related rates, considering the ladder as a right triangle formed by the ladder itself, the wall, and the ground.
Let's denote:
x: the distance from the base of the ladder to the house (in meters)
y: the height of the ladder on the wall (in meters)
θ: the angle between the ladder and the ground (in radians)
Given:
dx/dt = -2 m/sec (the rate at which the base of the ladder is moving away from the house)
Using the Pythagorean theorem, we have:
x^2 + y^2 = 5^2 (since the ladder has a length of 5 meters)
Taking the derivative of both sides with respect to time (t), we get:
2x(dx/dt) + 2y(dy/dt) = 0
(a) To find dy/dt:
We can solve the equation above for dy/dt:
2x(dx/dt) + 2y(dy/dt) = 0
2(3)(-2) + 2y(dy/dt) = 0 (substituting x = 3 and dx/dt = -2)
-12 + 2y(dy/dt) = 0
2y(dy/dt) = 12
dy/dt = 12/(2y)
dy/dt = 6/y
Now, we need to find y. Using the Pythagorean theorem again:
x^2 + y^2 = 5^2
3^2 + y^2 = 5^2
9 + y^2 = 25
y^2 = 25 - 9
y^2 = 16
y = 4 (taking the positive value as y represents a length)
Substituting y = 4 into dy/dt = 6/y:
dy/dt = 6/4
dy/dt = 3/2 m/sec
Therefore, the top of the ladder is sliding down the wall at a rate of 3/2 m/sec.
(b) To find dθ/dt:
We can use trigonometry to relate θ, x, and y:
tan(θ) = y/x
Differentiating both sides with respect to time (t), we get:
sec^2(θ)dθ/dt = (x(dy/dt) - y(dx/dt))/x^2
Substituting the given values:
sec^2(θ)dθ/dt = (3(3/2) - 4(-2))/3^2
sec^2(θ)dθ/dt = (9/2 + 8)/9
sec^2(θ)dθ/dt = (25/2)/9
sec^2(θ)dθ/dt = 25/18
Since sec^2(θ) is equal to 1 + tan^2(θ) and tan(θ) = y/x:
sec^2(θ) = 1 + (y/x)^2
sec^2(θ) = 1 + (4/3)^2
sec^2(θ) = 1 + 16/9
sec^2(θ) = (9 + 16)/9
sec^2(θ) = 25/9
Substituting sec^2(θ) = 25/9 into the equation:
(25/9)dθ/dt = 25/18
Simplifying and solving for dθ/dt:
dθ/dt = (25/18) * (9/25)
dθ/dt = 1/2 rad/sec
Therefore, (b) the angle between the ladder and the ground is changing at a rate of 1/2 rad/sec.
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There are 40 batteries in 10 packs.
Gavin wants to know how many batteries are in 1 pack.
Kylie wants to know how many batteries are in 5 packs.
Drag an expression to answer each question.
Help me Please I would appreciate your help!
:-):-):-):-):-)
Step-by-step explanation:
1) 40÷10
2) 40÷2
Tina drew a triangle with side lengths 6, 8, and 14. She claims it is a right triangle. Is she correct?
Answer:
No such a triangle can not exist.
Last year, a marketing research company estimated Amazon Prime members spent $1,825 on Amazon.com. The company believes that due to the pandemic. Amazon Prime members have spent more on average at Amazon.com compared to last year. They take a sample of 150 Amazon Prime members to test their belief. These sample members spent an average of $1,950. Assume the population standard deviation is $600.
Specify the hypotheses.
Null hypothesis (H0): µ ≤ 1825, the alternative hypothesis is µ > 1825.
The hypotheses are given below:
Null hypothesis (H0): µ ≤ 1825
Alternative hypothesis (Ha): µ > 1825 where µ represents the population mean amount spent by Amazon Prime members on Amazon.com.
As given in the question, the company believes that due to the pandemic, Amazon Prime members have spent more on average at Amazon.com compared to last year.
Therefore, the alternative hypothesis is µ > 1825.
The null hypothesis is that there is no significant difference or increases in the mean amount spent on Amazon.com by Amazon Prime members last year and this year or that the mean amount spent this year is less than or equal to that of last year, which is µ ≤ 1825.
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Find the probability using the normal distribution: P(0 P(0
The probabilities are:
P(0 < z < 1.96) = 0.4750
P(-1.23 < z < 0) = 0.3907
P(z > 0.82) = 0.2061
P(z < -1.77) = 0.0384
P(-0.20 < z < 1.56) = 0.5199
P(1.12 < z < 1.43) =0.0550
P(z > -1.43) = 0.9236
To locate the chances of the usage of the same old ordinary distribution, we will use a well-known normal desk or a calculator. Here are the calculations for the given possibilities:
P(0 < z < 1.96):
Using the standard everyday desk, the place to the left of one.Ninety-six is 0.9750, and the vicinity to the left of zero is 0.5000. Therefore, the chance between zero and 1.96 is:
P(0 < z < 1.96) = 0.9750 - 0.5000 = 0.4750
P(-1.23 < z < 0):
Using the usual ordinary table, the vicinity to the left of -1.23 is 0.1093, and the area to the left of zero is zero.5000. Therefore, the possibility between -1.23 and 0 is:
P(-1.23 < z < 0) = 0.5000 - 0.1093 = 0.3907
P(z > 0.82):
Using the standard everyday table, the region to the left of zero.82 is zero.7939. Therefore, the possibility of z being extra than 0.82 is:
P(z > 0.82) = 1 - 0.7939 = 0.2061
P(z < -1.77):
Using the same old regular desk, the vicinity to the left of -1.Seventy-seven is zero.0384. Therefore, the chance of z being much less than -1.77 is:
P(z < -1.77) = 0.0384
P(-zero.20 < z < 1.56):
Using the standard ordinary desk, the area to the left of -0.20 is 0.4207, and the area to the left of 1.56 is 0.9406. Therefore, the opportunity between -0.20 and 1. Fifty-six is:
P(-0.20 < z < 1.56) = 0.9406 - 0.4207 = 0.5199
P(1.12 < z < 1.43):
Using the standard regular desk, the place to the left of 1.12 is 0.8686, and the location to the left of one. Forty-three is zero.9236. Therefore, the opportunity between 1.12 and 1. Forty-three is:
P(1.12 < z < 1.43) = 0.9236 - 0.8686 = 0.0550
P(z > -1.43):
Using the same old normal desk, the location to the left of -1.43 is zero.0764. Therefore, the possibility of z being greater than -1. Forty-three is:
P(z > -1.43) = 1 - 0.0764 = 0.9236
Please word that the chances are rounded to 4 decimal locations for clarity.
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The correct question is:
"Find the probability using the standard normal distribution. P(0 < z <1.96), P(-1.23 < z < 0), P(z > 0.82), P(z < -1.77), P(-0.20 < z < 1.56), P(1.12 < z < 1.43), P(z > -1.43)"
Let p(x)=x+3x²-4x-12 a) Find the x-intercepts of the graph of p(x) c) Sketch a graph of p(x) b) Find the y-intercept of the graph of p(x)
a) The x-intercepts of p(x) are x = 3 and x = -4/3. b) The y-intercept of p(x) is y = -12.
a) To find the x-intercepts, we set p(x) = 0 and solve for x:
x + 3x² - 4x - 12 = 0
Simplifying the equation, we get:
3x² - 3x - 12 = 0
Factoring, we have:
(x - 3)(3x + 4) = 0
Setting each factor equal to zero, we find the x-intercepts:
x = 3 and x = -4/3
b) The y-intercept is the value of p(0), so we substitute x = 0 into the equation:
p(0) = 0 + 3(0)² - 4(0) - 12 = -12
Therefore, the y-intercept is -12.
c) To sketch the graph, we plot the x-intercepts (x = 3 and x = -4/3) and the y-intercept (y = -12) on a coordinate plane.
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Use the following diagram to answer question 4 and 5
Exterior Angle Theorem
8.6 points
The measure of ZA is
Given the following diagram, find the measures of
LA and ZB
Type your answer...
B (2x + 4)
5
8.6 points
The measure of ZB is
Type your answer...
(3x - 13)
116°
А
Answer:
3x-13+2x+4=116( the sum of two interior angle is epual to the sum of exterior angle )
5x =116-9
5x=107
x=21.4 degree
then
angle a= 3*21.4-13
a=51.2degree
angle b=2*21.4+2
b =44.8degree
Step-by-step explanation:
plz make me brainliest
I NEEED HELP match the equation with the term it belongs to. 3.14 *r *r. 2* 3.14 *r
Answer:
3.14 * r * r = area of circle, 2 * 3.14 * r = circumference
Step-by-step explanation:
Remember these formulas, very important.
plz help fast!!!!!!!!!!!!!!!!!!!!!
Answer:
D) 156/100 = n/70
Step-by-step explanation:
n = 156% of 70
% indicates a fraction over 100.
"of" indicates multiplication.
n = 156/100 • 70
Let's divide both sides by 70 to have one fraction one each side.
n/70 = 156/100
This is equal to D) [tex](\frac{156}{100}) = (\frac{n}{70})[/tex]
If a researcher wants to determine if there is a linear relationship between the number of hours a person goes without sleep and the number of mistakes he makes on a simple test. The following data is recorded.
n = 4, Σx = 20, Σy = 25, Σxy = 144 & Σx² = 120.
Find the equation of the regression line:
y = a + bx. y = (a = ____) + (b = _____)x.
b = [n(Σxy) - (Σx)(Σy)] / [n(Σx²) - (Σx)²] = ____
a = [(Σy)(Σx²) - (Σx)(Σxy)] / [n(Σx²) - (Σx)²] = _____
The equation of the regression line is: y = 1.5 + 0.95x
How to find the equation of the regression lineTo find the equation of the regression line, we can use the formulas:
b = [n(Σxy) - (Σx)(Σy)] / [n(Σx²) - (Σx)²]
a = [(Σy)(Σx²) - (Σx)(Σxy)] / [n(Σx²) - (Σx)²]
Given the following data:
n = 4
Σx = 20
Σy = 25
Σxy = 144
Σx² = 120
Let's calculate the values step by step:
b = [4(144) - (20)(25)] / [4(120) - (20)²]
= (576 - 500) / (480 - 400)
= 76 / 80
= 0.95
a = [(25)(120) - (20)(144)] / [4(120) - (20)²]
= (3000 - 2880) / (480 - 400)
= 120 / 80
= 1.5
Therefore, the equation of the regression line is:
y = 1.5 + 0.95x
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What is the smallest solution to the equation 3x2 - 16 = 131?
Answer:
7,-7 are the solutions
Step-by-step explanation:
Not a perfect square so its
+-7
I would go with -7 since its the lowest amount
Answer:
-7
Step-by-step explanation:
what is the volume of a right triangular prism whose height is 20 units and whose base is a right triangle with side lengths of 3,4, and 5?
What’s the opposite of -5/8
Find the measurements of DBC
Answer:
72 degrees
Step-by-step explanation:
To determine the value of DBC, we first have to determine the value of q
Angle on a straight line = 180 degrees
7q - 46 + 3q + 6 = 180
10q - 40 = 180
collect like terms
10q = 180 + 40
10q = 220
q = 22
Substitute for q in angle dbc
3(22) + 6 = 72 degrees
HELP PWEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer:
rap music
Step-by-step explanation:
^
Answer:
i’m not sure if these are right or not but
Step-by-step explanation:
the first one is no
i think the second one is no
i think the third one is yes
i think the fourth one is yes
i am not sure and i’m REALLY sorry if its wrong
Use Strong Induction to prove that: If p + 1/p E N, then Pn+1/Pn EN for all nEN.
We prove with the help of Strong Induction.
Let P(n) be the statement that Pn+1/Pn E N.
In order to prove this statement, we will utilize strong induction.So we are given that p + 1/p E N. We will show that P(n) is true for all n >= 1.
Let's consider the base case
P(1):P2/P1 = (p + 1/p)^2 - 2 = (p^2 + 2 + 1/p^2) - 2p/p = (p^2 + 1/p^2) - (2p - 2/p)
Since p + 1/p E N, both p and 1/p must be integers.
Hence, p^2 and 1/p^2 are also integers. This implies that (p^2 + 1/p^2) is an integer.
It only remains to show that (2p - 2/p) is an integer. This is equivalent to showing that 2p^2 - 2 E 0 mod p. But this is clearly true, since 2p^2 - 2 = 2(p^2 - 1) and p^2 - 1 is divisible by p.
Let's assume that P(k) is true for all k such that 1 <= k <= n. We need to prove that P(n+1) is true as well.
Now we need to prove that P(n+1) is true. In other words, we need to show that P(n+2)/P(n+1) E N, assuming that P(n+1)/P(n) E N and P(n)/P(n-1) E N.
Using the definition of P(n), we have:P(n+1)/P(n) E N and P(n)/P(n-1) E N imply that P(n+1) = aP(n) and P(n) = bP(n-1) for some integers a and b. Then:P(n+2)/P(n+1) = (P(n+2)/P(n+1)) * (P(n)/P(n)) = (P(n+2)P(n))/(P(n+1)P(n)) = (aP(n+1)P(n))/(bP(n)P(n+1)) = a/bIf we can show that a/b E N, then P(n+2)/P(n+1) E N, and P(n+1) satisfies the inductive hypothesis.
But this follows from the fact that a and b are integers and the product of two integers is always an integer.
Hence, P(n+1) is true for all n >= 1, by strong induction.Therefore, by strong induction, we have proved that if p + 1/p E N, then Pn+1/Pn EN for all nEN.
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Wiseman Video plans to make four annual deposits of $2,000 each to a special building fund. The fund’s assets will be invested in mortgage instruments expected to pay interest at 12% on the fund’s balance. (FV of $1, PV of $1, FVA of $1, PVA of $1, FVAD of $1 and PVAD of $1) (Use appropriate factor(s) from the tables provided.)
Using the appropriate annuity table, determine how much will be accumulated in the fund on December 31, 2019, under each of the following situations.
1. The first deposit is made on December 31, 2016, and interest is compounded annually.
Table or calculator function: FVA of $1
Payment: $2,000
n = 4
i = 12%
Fund balance 12/31/2019: $9,559
2. The first deposit is made on December 31, 2015, and interest is compounded annually.
Table or calculator function: FVAD of $1
Payment: $2,000
n = 4
i = 12%
Fund balance 12/31/2019: $10,706
3. The first deposit is made on December 31, 2015, and interest is compounded quarterly.
Using the FV of $1 chart, calculate the fund balance:
Deposit Date i = n = Deposit Fund Balance 12/31/2019
12/31/2015 3% 16 $2,000 $3,209
12/31/2016 3% 12 2,000 2,852
12/31/2017 3% 8 2,000 2,534
12/31/2018 3% 4 2,000 2,251
$10,846
4. The first deposit is made on December 31, 2015, interest is compounded annually, and interest earned is withdrawn at the end of each year.
Deposit Amount No. of Payments Interest left in Fund Fund Balance 12/31/2019
$2,000 $8,000
The fund balance at the end of 2019 will be $8,000.
The given problem has four different parts, where we are supposed to calculate the accumulation of funds at the end of 2019 in different scenarios.
Scenario 1In the first scenario, the first deposit is made on December 31, 2016, and interest is compounded annually.
Using the FVA of $1 table; Payment: $2,000n = 4i = 12%
Fund balance 12/31/2019: $9,559
Hence, the fund balance at the end of 2019 will be $9,559.Scenario 2In the second scenario, the first deposit is made on December 31, 2015, and interest is compounded annually.
Using the FVAD of $1 table;Payment: $2,000n = 4i = 12%
Fund balance 12/31/2019: $10,706 Therefore, the fund balance at the end of 2019 will be $10,706.Scenario 3In the third scenario, the first deposit is made on December 31, 2015, and interest is compounded quarterly. Using the FV of the $1 chart, we get the following calculation:
Deposit Date i = n = Deposit Fund Balance 12/31/2015 3% 16 $2,000 $3,20912/31/2016 3% 12 $2,000 $2,85212/31/2017 3% 8 $2,000 $2,53412/31/2018 3% 4 $2,000 $2,251
The interest rate is 3%, and the payment is $2,000. Hence, the fund balance at the end of 2019 will be $10,846.Scenario 4In the fourth scenario, the first deposit is made on December 31, 2015, interest is compounded annually, and interest earned is withdrawn at the end of each year.
Deposit Amount No. of Payments Interest left in Fund Fund Balance 12/31/2019$2,000 $8,000 Hence, the fund balance at the end of 2019 will be $8,000.
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Solve each inequality.
2s+5> 49
_
Answer: S > 22
Step-by-step explanation:
Answer:
s = 23 and above
Step-by-step explanation:
2s + 5 > 49
-5 -5
2s > 44
44/2
= 22
However, the answer for s is not 22
Since it has this sign > not ≥ then that means s = 23 and above
Check:
2(23) + 5 > 49
46 + 5 > 49
51 > 49
Write an expression in factored form to represent the area of each shaded region. Include the area formula and full steps and write your answer in simplified form. 3x a) (3 marks) 3x + y 2x +y 3x b) 3y (3 marks) B 2x 2x
a. The area of the shaded region = (3x)(5x + 2y).
b. The area of shaded region = 2x(2xπ - 3x).
Given that,
We have to write an expression in factored form to represent the area of each shaded region.
We know that,
a. In the picture we can see that the shaded region.
The area of the shaded region is area of rectangle with vertical + area of rectangle Horizontal
The area of rectangle is length × width.
The area of the shaded region = (3x)(3x + y) + (2x + y)(3x)
The area of the shaded region = (3x)(3x + y + 2x + y)
The area of the shaded region = (3x)(5x + 2y)
Therefore, The area of the shaded region = (3x)(5x + 2y)
b. In the picture we can see that circle with shaded region,
The area of shaded region is area of the circle - area of the rectangle
The area of the circle is πr² and area of rectangle is length × width.
The area of shaded region = πr² - (l×w)
The area of shaded region = π(2x)² - (2x × 3x)
The area of shaded region = 4x²π - 6x²
The area of shaded region = 2x(2xπ - 3x)
Therefore, The area of shaded region = 2x(2xπ - 3x).
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Let be an equivalence relation on a set S, and let a, b e S. Show that two equivalence classes under ~ are either equal or disjoint, i.e. either [a] = [b] or [a] n [b] = 0.
Given, an equivalence relation ~ on a set S. Let a and b be two elements in the set S. Assuming that [a] and [b] are two equivalence classes under the equivalence relation ~. Now we need to prove that either [a] = [b] or [a] ∩ [b] = ∅ (disjoint).
Proof:If [a] and [b] are not equal, then there must be some element c in the intersection of the equivalence classes [a] and [b]. i.e, c belongs to [a] and c belongs to [b].Thus, [a] ∩ [b] is not empty.
Let x be an element in [a], then x~a, and a~c (since c belongs to [a]) and hence x~c. So, x belongs to [c] which implies that [a] is a subset of [c].Now, let y be an element in [b], then y~b, and b~c (since c belongs to [b]) and hence y~c. So, y belongs to [c] which implies that [b] is a subset of [c].Thus, both [a] and [b] are subsets of [c].
Therefore, if [a] and [b] are not equal, then [a] and [b] are both subsets of [c] and hence the intersection of [a] and [b] is not empty. Thus, [a] and [b] are not disjoint. Hence, the proof by contradiction.
Conversely, if [a] and [b] are disjoint, then [a] ∩ [b] = ∅. And thus, [a] is not equal to [b].Therefore, two equivalence classes under the equivalence relation ~ are either equal or disjoint.
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Make r the subject of x=e+r/d
Answer:
r = dx - de
Step-by-step explanation:
x = e + r/d
x - e = r/d
d (x - e) = r
dx- de = r
r = dx - de