Cantor's Theorem states that the cardinality of a set A is strictly less than the cardinality of its power set P(A), for every set A. In other words, there is no bijection between A and P(A).
The proof of Cantor's Theorem relies on a diagonalization argument. Suppose there is a bijection f between A and P(A). We can use f to construct a subset B of A that is not in the image of f.
To do this, we define B as follows: for each element x in A, if x is not in the set f(x), then we add x to B. In other words, B contains all elements of A that are not in their corresponding set in P(A) under f.
Now, we show that B is not in the image of f. Suppose that there exists some element y in A such that f(y) = B. Then, we have two cases: either y is in B or y is not in B.
If y is in B, then y is not in f(y), since y was added to B precisely because it is not in its corresponding set in P(A) under f. But this contradicts the assumption that f(y) = B.
If y is not in B, then y is in f(y), since y is not in B precisely because it is in its corresponding set in P(A) under f. But this also contradicts the assumption that f(y) = B.
Therefore, we have shown that B is not in the image of f, which contradicts the assumption that f is a bijection between A and P(A). Thus, there can be no such bijection, and Cantor's Theorem follows.
The consequence of Cantor's Theorem is that there are different sizes of infinity, which has profound implications for mathematics and philosophy. It shows that there are sets that are "larger" than others, and that there is no "largest" infinity. This has led to the development of set theory as a foundational branch of mathematics, and has influenced debates about the nature of infinity in philosophy.
The 4th term of an arithmetic sequence is 24 and the 12th term is 56, what is the first term?
Answer:
First term = 12
Step-by-step explanation:
A4 = 24
A12 = 56
An = A1 + (n-1)d; this defines the formula of an arithmetic series
A4 = 24 = A1 + (4-1)d
A4 = 24 = A1 + 3d
A12 = 56 = A1 + (12-1)d
A12 = 56 = A1 + 11d
Subtract A4 from A12 to get d (distance)
(56 = A1 + 11d) - (24 = A1 +3d)
32 = 8d
d = 4
Substitute d = 4 into A4
A4 = 24 = A1 + 3(d)
24 = A1 + 3(4)
24 = A1 + 12
24 - 12 = A1
12 = A1
Write a negative integer and a positive integer whose sum is –5.
Answer:
-18+ 13
-6 + 1
-20 + 15
-7 + 2
Step-by-step explanation:
There are some hens and some sheeps are in a farm,total number of animals are 10 and total number of their legs are 32 1.Find number of hens? 2. Find number of sheeps?
Answer: The number of hens are 4 and number of sheeps are 6
Step-by-step explanation:
Given : Total animals = 10
Total number of legs = 32
Let number of hens = h and number of sheep = s
Thus h+s = 10 (1)
Now each hen has two legs and each sheep has 4 legs :[tex]2\times h+4\times s=32[/tex] (2)
Solving for h and s we get:
s = 6 , h =4
Thus number of hens are 4 and number of sheeps are 6
Pre-image ABCD was dilated to produce image A'B'C'D' What is the scale factor from the pre-image to the image?
Answer:
The answer is 3/4
Step-by-step explanation:
Can someone help me with this. Will Mark brainliest.
Answer:
[tex]\sqrt{53}[/tex]
Step-by-step explanation:
Use the distance formula:
[tex]\sqrt{(x1-x2)^2+(y1-y2)^2}[/tex]
plug in (-5,6) for (x1,y1) and (-3,-1) for (x2,y2)
[tex]\sqrt{(-5-(-3))^2+(6-(-1))^2}[/tex]
[tex]\sqrt{4+49}[/tex]
[tex]\sqrt{53}[/tex]
Hope that helps :)
At LaGuardia Airport for a certain nightly flight, the probability that it will rain is
0.09 and the probability that the flight will be delayed is 0.18. The probability that it
will rain and the flight will be delayed is 0.04. What is the probability that the flight
would be delayed when it is not raining? Round your answer to the nearest
thousandth.
Answer:
A, The probability that it will rain is 0.09 and the probability that the flight will be delayed is 0.18.
Step-by-step explanation:
Answer: 0.154
Step-by-step explanation:
Q2 Solve the following initial value problem 1 y" + 4y = r - sin 3x y(0) = 1, (0) by using method of undetermined coefficients. (10 marks)
The given initial value problem using the method of undetermined coefficients. The final solution to the initial value problem is y = cos(2x) + x - (1/9)*sin(3x).
To solve the given initial value problem, we begin by finding the general solution to the associated homogeneous equation. The homogeneous equation is given by y'' + 4y = 0. The characteristic equation is obtained by substituting y = e^(mx) into the equation, resulting in the quadratic equation m^2 + 4 = 0. Solving this equation yields two distinct roots: m_1 = 2i and m_2 = -2i. Thus, the general solution to the homogeneous equation is y_h = c_1cos(2x) + c_2sin(2x), where c_1 and c_2 are arbitrary constants.
Next, we assume a particular solution in the form of y_p = Ax + B + Csin(3x) + Dcos(3x), where A, B, C, and D are undetermined coefficients. We substitute this particular solution into the given differential equation and solve for the coefficients. Comparing the coefficients of like terms, we find A = 0, B = 1, C = -1/9, and D = 0.
The particular solution is y_p = x - (1/9)*sin(3x), and the complete solution is obtained by adding the particular solution to the homogeneous solution: y = y_h + y_p. Applying the initial condition y(0) = 1, we find that c_1 = 1 and c_2 = 0. Therefore, the final solution to the initial value problem is y = cos(2x) + x - (1/9)*sin(3x).
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A sample of data with n = 66 observations is used to run a multiple regression with 7 independent variables.
If this data set has SST = 2326.9523 and SSR = 1632.8843, what is the value of the F-statistic for the test of overall significance of this regression relationship?
The value of the F-statistic for the test of overall significance of this regression relationship is approximately 11.245.
Given:SST = 2326.9523 SSR = 1632.8843 Independent variables = 7
Observations = 66Formula used:F = (SSR/K) / (SSE / (n-K-1))
Where SSR = Regression sum of squares SSE = Error sum of squaresK = Number of independent variables n = Number of observations
To calculate the F-statistic, we first need to calculate SSE which is given by:SSE = SST - SSR = 2326.9523 - 1632.8843 = 693.068
The degrees of freedom for the regression are K and the degrees of freedom for the error term are n - K - 1 = 66 - 7 - 1 = 58
The value of the F-statistic is given by:F = (SSR/K) / (SSE / (n-K-1))= (1632.8843/7) / (693.068 / 58)= 11.245
Therefore, the value of the F-statistic for the test of overall significance of this regression relationship is approximately 11.245.
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T=21and u =4 what is \sqrt(t+U)
Answer: 5
Step-by-step explanation: 21 + 4 = 25
square root of 25 = 5
Answer and Step-by-step explanation:
T = 21
U = 4
[tex]\sqrt{ 21 + 4} \\\\\\\sqrt{25} \\\\\\5[/tex]
5 is the answer to the expression.
#teamtrees #PAW (Plant And Water)
What is the difference between 3/4 and one
Answer:
3/4 is 0.75 and 1 is 1
Step-by-step explanation:
:\
hey what's the answer ?
[tex] \sqrt{36 \times 2 \times 2 \times 3 \times 3} [/tex]
Step-by-step explanation:
36 is your answer
I hope.it helps mate
have a nice day
#Captainpower
a toy rocket is fired into the air from the top of a barn its height H above the ground in yards after T seconds is given by the function h of T is equal to -5 t^2 + 10t + 20
1.what is the maximum height of the rocket
2. how long was the rocket in the air before hitting the ground
3.at what time(s) will the rocket be at a height of 22 yards
Answer:
45
Step-by-step explanation:
x = -30/2(-5)
x = -30/(-10)
x = 3 sec
.
Plug above into equation to find height:
h=-5t^2 + 30t
h=-5*3^2 + 30(3)
h=-5*9 + 30(3)
h=-45 + 90
h=45
Find the total amount owed, to the nearest cent, for the following simple interest loans.
(Step-by-step explanation)
a. $525 loan at 9.9% interest for 6 months
b. $12,460 loan at 5.6% interest for 30 months
a. The total amount owed for the $525 loan at 9.9% interest for 6 months is approximately $556.19 when rounded to the nearest cent.
b. The total amount owed for the $12,460 loan at 5.6% interest for 30 months is approximately $33,504.80 when rounded to the nearest cent.
a. To calculate the total amount owed for the $525 loan at 9.9% interest for 6 months, we can use the formula for simple interest:
Total amount owed = Principal + (Principal × Interest Rate × Time)
Given:
Principal (P) = $525
Interest Rate (R) = 9.9% = 0.099 (converted to decimal)
Time (T) = 6 months
Plugging these values into the formula, we get:
Total amount owed = $525 + ($525 × 0.099 × 6)
Simplifying the equation:
Total amount owed = $525 + ($31.185)
Total amount owed = $556.185
Therefore, the total amount owed for the $525 loan at 9.9% interest for 6 months is approximately $556.19 when rounded to the nearest cent.
b. To calculate the total amount owed for the $12,460 loan at 5.6% interest for 30 months, we'll follow the same formula for simple interest:
Total amount owed = Principal + (Principal * Interest Rate * Time)
Given:
Principal (P) = $12,460
Interest Rate (R) = 5.6% = 0.056 (converted to decimal)
Time (T) = 30 months
Plugging these values into the formula, we get:
Total amount owed = $12,460 + ($12,460 * 0.056 * 30)
Simplifying the equation:
Total amount owed = $12,460 + ($21,044.8)
Total amount owed = $33,504.8
Therefore, the total amount owed for the $12,460 loan at 5.6% interest for 30 months is approximately $33,504.80 when rounded to the nearest cent.
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plzzzz help me plzzzzz z z z zz z !!!!!!!!!
Answer:
circle: ( 1, 0.5 )
square: ( 4.5, 3.5 )
triangle: ( 0, 2.5 )
Step-by-step explanation:
What is the answer to this question?
Answer:
what should i answer?
Step-by-step explanation:
Ay benefit concert, twelve bands have volunteered to perform but there is only enough time for eight of the bands to play. How many lineups are possible?
The requried, there are 495 possible combination lineups with eight bands out of twelve for the benefit concert.
To determine the number of possible lineups with eight out of twelve bands performing, we need to calculate the combination.
The number of ways to choose eight bands out of twelve without considering the order is given by the combination formula:
[tex]C(n, r) = \dfrac{n!}{ (r! * (n-r)!)}[/tex]
where n is the total number of bands (12 in this case) and r is the number of bands to be chosen (8 in this case).
Let's calculate it:
[tex]C(12, 8) =\dfrac {12!}{(8! * (12-8)!)}[/tex]
[tex]= \dfrac{12!} {(8! * 4!)}[/tex]
= (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1)
= 495
Thus, there are 495 possible combination lineups with eight bands out of twelve for the benefit concert.
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(50 POINTS) Express each sum using summation notation.
14. 3 + 3^2/2 + 3^3/3 ... + 3^n/n
15. 1 + 3 + 5 + 7 +... [2(12) - 1]
I'm on my school computer so the picture is blocked sorry :(
Answer:
the answer is in the photo hope it help
A rectangular pool has an area of 50 square feet. What are the lengths of its sides? There is more than one possible answer. Draw a picture of the possible answers.
Answer:
10*5 or 25*2
Step-by-step explanation:
There are other answers but those involve decimals and I'm assuming they want whole numbers.
nCk means the number of ways we can choose k objects from n objects
Find:
∑10k=010Ck
The sum of the binomial coefficients nCk, where n ranges from 0 to 10 and k varies from 0 to 10, is equal to [tex]2^10[/tex], which is 1024.
The expression ∑10k=010Ck represents the sum of the binomial coefficients for all possible values of k from 0 to 10. The binomial coefficient nCk, also known as "n choose k," represents the number of ways we can choose k objects from a set of n objects.
In this case, we are summing up the binomial coefficients for n ranging from 0 to 10. For each value of n, we calculate the binomial coefficient nCk for k values ranging from 0 to 10. The formula to calculate the binomial coefficient is n! / (k!(n-k)!), where "!" denotes the factorial operation.
When we substitute the values into the formula, we find that all the binomial coefficients are 1, except when k equals 0 or n. In these cases, the binomial coefficient is equal to 1, as there is only one way to choose 0 objects or all n objects from a set.
Since there are 11 values of n (0 to 10), and for each n there is one non-zero binomial coefficient, the sum of all the binomial coefficients from 0 to 10 is equal to 11. Therefore, the expression simplifies to ∑10k=010Ck = 11.
So, the sum of all the binomial coefficients ∑10k=010Ck is equal to 11, which means we have 11 ways to choose k objects from a set of 10 objects. Another way to interpret this is that the sum of the binomial coefficients represents the number of subsets of a set with 10 elements, which is 11. However, if we are considering the case of choosing 0 to 10 objects, the sum becomes 2^10, which is equal to 1024.
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Which of the following is true?
Answer:
option B should be correct
sound travels at 320.29 meters per second at sea level. how many miles will sound travel at sea level in 7 seconds?
round off the answer to the nearest thousandths
thx in advance
Answer:
1.393 milesExplanation:
320.29 × 7 = 2,242.03 m = 1.39313285413187 miles ≈ 1.393
If c ⊥ b and a || c, what is m∠2?
Answer:
A) 90°
Step-by-step explanation:
if lines b and c are perpendicular then all 8 angles measure 90 degrees
Find the LCM and HCF of 64 and 72 using prime factors
What is the LCM of 64 and 72?
Find the prime factorization of 64.
Find the prime factorization of 72. 72 = 2 × 2 × 2 × 3 × 3.
Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the LCM: LCM = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3.
LCM = 576.
What is the GCF of 64 and 72?
Find the prime factorization of 64. 64 = 2 × 2 × 2 × 2 × 2 × 2.
Find the prime factorization of 72. 72 = 2 × 2 × 2 × 3 × 3.
To find the GCF, multiply all the prime factors common to both numbers: Therefore, GCF = 2 × 2 × 2.
GCF = 8.
Answer:
8x8 and 8x9
Step-by-step explanation:
because 8x8 is 64 and 8x9 is 72
A five-year project that will require $3,200,000 for new fixed assets will be depreciated straight-line to a zero book value over six years. At the end of the project, the fixed assets can be sold for $640,000. The tax rate is 32% and the required rate of return is 13.30%. What is the amount of the aftertax salvage value?
As per the given values, the after-tax salvage value is $435,200.
Amount required = $3,200,000
Time = 6 years
Calculating the accumulated depreciation -
Amount/ Number of years
= $3,200,000 / 6
= $533,333.3.
Calculating the accumulated depreciation at project end -
= 6 x $533,333.33
= $3,200,000.
Calculating the book value of the fixed assets -
Book value = Cost of fixed assets - Accumulated depreciation
= $3,200,000 - $3,200,000
= $0
Calculating the taxable gain or loss on the sale of the fixed assets -
Taxable gain/loss = Selling price - Book value
= $640,000 - $0
= $640,000
Calculating the tax liability -
Tax liability = Tax rate x Taxable gain
= 0.32 x $640,000
= $204,800
Calculating the after-tax salvage value -
After-tax salvage value = Selling price - Tax liability
= $640,000 - $204,800
= $435,200
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Plz help been stuck for a while now
Describe how to find the sale price of an item that has been discounted 15%.
Multiply the original price by
% to find the sale price.
Multiply the discount percent to the marked price and subtract from the marked price.
What will be the selling price?As we know that the discount is always given on the marked price of the product.
We will take one example to understand it supposes the marked price of any object is MP= 100 and the discount =15% so to find the SP
First, find the discount amount
[tex]MP\times Discout \ percent =100\times \dfrac{15}{100} =15[/tex]
Now to find the SP subtract the discount from the MP
[tex]SP=MP-Discount[/tex]
[tex]SP=100-15=75[/tex]
So for finding SP multiply the discount percent to the marked price and subtract from the marked price.
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Mrs. Avery is going to randomly select one student from her class to read a poem out loud. There are 15 boys and 13 girls in her class
The probability of selecting a girl to read the poem out loud in her class is 46.4% .
To calculate the probability of selecting a girl to read a poem out loud in a class of 28 students, we can use the formula:
Probability = Number of Desirable Outcomes / Total Number of Outcomes
Here, the desirable outcome is selecting a girl to read the poem, and the total outcomes are all the students in the class.
Number of Desirable Outcomes:
Mrs. Avery has 13 girls in her class, so there are 13 desirable outcomes.
Total Number of Outcomes:
Mrs. Avery has 28 students in her class, so there are 28 total outcomes.
Probability = Number of Desirable Outcomes / Total Number of Outcomes
Probability = 13 / 28
Probability = 0.464 or 46.4%
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Note: The complete question is -Mrs. Avery is planning to have one student read a poem out loud in class. She has a class of 28 students, consisting of 15 boys and 13 girls. What is the probability that she will select a girl to read the poem out loud?
Point E is located at (-8,7). Point F is located at (9,7).
What is the distance, in units, between point E and point F?
Answer:
18 units to the right.
Step-by-step explanation:
Hope this helps!
A half-century ago, the mean height of women in a particular country in their 20s was 62.9 inches. Assume that the heights of today's women in their 20s are approximately normally distributed with a standard deviation of 2.87 inches.
If the mean height today is the same as that of a half-century ago, what percentage of all samples of 21 of today's women in their 20s have mean heights of at least 64.76 inches?
About _____ % of all samples have mean heights of at least 64.76 inches.
Approximately 0.11 percent of all samples have mean heights of at least 64.76 inches.
Given: 50 years prior, the mean level of ladies in a specific country in their 20s was 62.9 inches. With a standard deviation of 2.87 inches, women in their 20s today have heights that are roughly typical. We must determine the proportion of all 21 samples that have mean heights of at least 64.76 inches among today's women in their 20s. Bit by bit clarification:
Let μ be the mean level of ladies in this day and age and allow n to be the example size. We can find the likelihood of an example mean being more prominent than or equivalent to 64.76 inches utilizing the z-score recipe, as follows: z = (x - μ)/(σ/√n) = (64.76 - 62.9)/(2.87/√21) = 3.06The likelihood that an example of 21 ladies will have a mean level more noteworthy than or equivalent to 64.76 inches is equivalent to the likelihood that a typical irregular variable Z is more noteworthy than or equivalent to 3.06.
This probability is approximately 0.0011, or 0.11 percent, according to a typical normal table or calculator. This means that approximately 0.11 percent of all samples have mean heights of at least 64.76 inches.
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You estimate that there are 56 marbles in a jar. The actual amount is 70 marbles. Find the percent error.
Answer:
80 percent of 70 is 56, so you were 20% wrong?
The percent error is 20%
let me know if you need anything else :)