Given that you order one donut and you know that it is not the blueberry one. We need to determine the probability that it is the chocolate one.
Say we have 3 flavors of donuts such as Blueberry, Chocolate and Glazed. There are a total of 3 possibilities for the first choice. If we assume that the person randomly selects one of these donuts, each donut is equally likely to be chosen.
Here, The donut selected is not the blueberry one and we need to find the probability that it is the chocolate one. Number of blueberry donuts = 1Probability of choosing the blueberry donut = 1/3 (Since there are 3 possible donuts and only 1 of them is the blueberry one) Therefore, the probability of choosing a chocolate donut is 1 - 1/3 = 2/3.
Similarly, the probability of choosing a glazed donut is also 2/3. Thus, the probability of choosing a chocolate donut or a glazed donut is 2/3. Answer: 2/3.
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Patrick left an $7 tip on a $53 restaurant bill. What percent tip is that?
The percent tip is 13.2%.
To find out what percentage Patrick gave as tip, we need to divide the tip amount by the total bill amount and then convert the result into a percentage.
For this question:
Patrick left an $7 tip on a $53 restaurant bill. What percent tip is that?
Solution:The percentage of tip can be found using the following formula:
% = (Tip amount / Total bill amount) x 100
Plugging in the given values we get:
% = (7 / 53) x 100% = 0.132 x 100% = 13.2 %
Therefore, Patrick gave a 13.2% tip on his restaurant bill.
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Assuming an individual has X hours per week during the summer to devote to taking classes or working for an employer. Also, assume that to receive at least a B or better, you must devote at least X hours per course per week (1 course = X hours per week, 2 courses = X hours per week, etc...) Explain why one faces a tradeoff between the number of courses and hours of work. Why would it be impossible to take X courses while working X hours per week? Why would taking one class and working X hours per week would result in large amount of free time?
One faces a tradeoff between the number of courses and hours of work due to limited available time. Taking X courses while working X hours per week is impossible as it exceeds the time constraint.
The tradeoff between the number of courses and hours of work arises due to the finite number of hours available in a week. Assuming an individual has X hours per week, this time must be divided between courses and work.
To achieve a B or better in a course, it is necessary to dedicate at least X hours per course per week. Therefore, taking X courses while working X hours per week would require dedicating X hours per course, resulting in a total time commitment exceeding the available X hours.
Conversely, if only one class is taken while working X hours per week, the time commitment for a single course is less than X hours. This scenario leaves a significant amount of free time remaining, as the total time dedicated to the course and work does not exhaust the available X hours.
In summary, the tradeoff occurs because the time available is limited, and the minimum time requirement per course restricts the number of courses that can be taken while maintaining a specific work schedule.
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A rectangle with area 36 square inches has a length that is 3 less than 3 times the width. Find the length and the width of the rectangle.
Answer:
length = 9 in ; width = 4 in
Step-by-step explanation:
1) formula of the area of a rectangle
A = base x width
2) write an equation with the given informations:
w x (3w-3) = 36
3) solve the equation
- multiply W by 3w and -3
3w^2 -3w = 36
- divide the two members by 3
w^2 - w = 12
- add up -12 at the two members
w^2-w -12 = 12 -12
w^2 - w - 12 = 0
- find two numbers whose sum is -1 and whose product is -12 and factorise the equation
(w-4)(w+3) = 0
- find the two solutions
w-4 = 0
w = 4
w+3 = 0
w = -3
4)remember that a length can’t be negative.
w = 4 is the only solution
5) find the length
4x3 - 3 = 9
A stick of length 10 is broken at a point X which is uniformy distributed on (0,10). Given X = 1, another breakpoint Y is chosen uniformly on (0,3). The joint part(X,Y) is given by f(x,y) for 0 <<<10 10s The marginal pdf of Y is given by fY() = 0.1 for Oy10
The joint probability density function (pdf) is f(x,y) = 0.03 for 0 < x < 1 and 0 < y < 3.
The joint pdf, f(x,y), represents the probability density function for the random variables X and Y. Given that X is uniformly distributed on (0,10), we have fX(x) = 0.1 for 0 < x < 10. The probability of X being less than 1 is 1/10, so the conditional pdf f(x|X<1) = 0.1 for 0 < x < 1.
Furthermore, Y is uniformly distributed on (0,3), so fY(y) = 0.1 for 0 < y < 3. To find the joint pdf, we multiply the conditional pdf of X with the marginal pdf of Y: f(x,y) = f(x|X<1) * fY(y) = 0.1 * 0.1 = 0.01 for 0 < x < 1 and 0 < y < 3. Therefore, the joint pdf is f(x,y) = 0.01 for 0 < x < 1 and 0 < y < 3.
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Survey / Statistical Question:
How many times has each person moved states?
Answer:
a regular person would gave moved 4 to 5 times
Item 7
A flat rate shipping box is in the shape of a rectangular prism. You estimate that the volume of the box is 350 cubic inches. You measure the box and find that it has a length of 10 inches, a width of 9 inches, and a height of 4.5 inches. Find the percent error. Round your answer to the nearest tenth of a percent.
Answer: the answer is 6.06%, however if it is rounded it's 6.0%
Step-by-step explanation:
What is the area of the shaded area shown, his lawn? Please help me I really need it
Answer:
c - 13,600ft^2
Step-by-step explanation:
200 x 80 = 16,000ft^2
40 x 60 = 2,400ft^2
16,000ft^2 - 2,400ft^2 = 13,600ft^2
Add the negatives.
6- 3 + 2 - 1 - 4
Please help ....
Select the correct answer.
If the parent function RX) = x2 is modified to g(x) = 2x + 1, which statement is true about g(x)?
ΟΑ. .
It is an even function.
ОВ.
It is an odd function,
OC. It is both an even and an odd function.
OD.
It is neither an even nor an odd function.
For which equation is s = 6 not the solution?
4 s = 24
9 - s = 4
s + 6 = 12
s - 4 = 2
Answer:
9 - s = 4
Step-by-step explanation:
9 - 6 is 3, not 4, so thats the answer!
The equation is s = 6 not the solution is 9 - s = 4.
What is the equation?An equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.
For which equation is s = 6 not the solution?
1. The solution of the equation 4 s = 24 is;
[tex]\rm4 s = 24\\\\s=\dfrac{24}{4}\\\\s=6[/tex]
2. The solution of the equation 9 - s = 4 is;
[tex]\rm 9 - s = 4\\\\s=9-4\\\\s=5[/tex]
3. The solution of the equation s + 6 = 12 is;
[tex]\rm s + 6 = 12\\\\s=12-6\\\\s=6[/tex]
4. The solution of the equation s - 4 = 2 is;
[tex]\rm s - 4 = 2\\\\s=2+4\\\\s=6[/tex]
Hence, the equation is s = 6 not the solution is 9 - s = 4.
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Current Attempt in Progress The following table lists the ages (in years) and the prices (in thousands of dollars) by a sample of six houses. Age Price 27 165 15 182 3 205 35 177 9 180 18 161 The null hypothesis is that the slope of the population regression line of price on age is zero and the alternative hypothesis is that the slope of this population regression line is less than zero. The significance level is 5%. What is the value of the test statistic, t. rounded to three decimal places?
Whwn the null hypothesis is that the slope of the population regression line of price on age is zero, the test statistic is -2.43.
How to calculate the valueThe test statistic is calculated as follows:
t = (b - 0) / SE(b)
In this case, the slope of the regression line is estimated to be -11.50, the standard error of the slope is 4.77, and the hypothesized value of the slope is 0.
t = (-11.50 - 0) / 4.77
= -2.43
The test statistic is -2.43, The p-value for this test statistic can be calculated using a t-table. The degrees of freedom for this test are 5 - 2 = 3. The p-value for a t-statistic of -2.43 with 3 degrees of freedom is 0.048.
Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.
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Eduardo bought 2-liter sports drink for soccer practice.During practice,Eduardo drank 250 millimeters of the sports drink.Then he drank another 1/2 liter of the drink after practice.How many milliters of the sports drink did Eduardo have left
Answer:
1,250 millileters
Step-by-step explanation:
The computation of the number of milliters of the sports drink left is shown below:
Given that
The 2-liter sports drink is purchased
And, drank 250 millimeters and than drank one-half liter of the drink
Now convert liter to millileter
1 litre = 1000 milliliter.
2 liters = 2 × 1000 milliliters.
= 2000 milliliters.
Now the left drink would be
= (2000 - 250) - (250 × 2)
= 1,750 - 500
= 1,250 millileters
In a large population, 92% of the households have cable tv. A simple random sample of 225 households is to be contacted and the sample proportion computed. What is the mean and standard deviation of the sampling distribution of the sample proportions
Answer:
the mean and the standard deviation is 0.92 and 0.01808 respectively
Step-by-step explanation:
The computation of the mean and the standard deviation is shown below:
The mean is 0.92
And, the standard deviation is
= √0.92 × (1 - 0.92) ÷ √225
= √0.92 × 0.08 ÷ √225
= √0.0736 ÷ √225
= √3.27
= 0.01808
Hence, the mean and the standard deviation is 0.92 and 0.01808 respectively
If a dog can eat 16 treats in 2 hours, how many can it eat in 6 hours?
Answer:
48 treats
Step-by-step explanation:
16 treats / 2 hours = 8 treats / hour
8 treats / hour x 6 hours =
48 treats
Hope that helps! :)
A coin is tossed four times. You bet $1 that heads will come up on all four tosses. If this happens, you win $10. Otherwise, you lose your $1 bet.
Find: P(you win) =
P(you lose) =
Average winnings, µ, =
The P(you win) = 1/16P(you lose) = 15/16 and Average winnings, µ, = -5/16.
The probability of winning (P) and the probability of losing (Q) are both possible outcomes from a coin toss experiment. The probability of winning is given by the formula
P = Number of ways to win/ Total number of possible outcomes.The probability of losing is given by the formula
Q = Number of ways to lose/Total number of possible outcomes.
Formulae used to calculate probability are:
P = Number of ways to win/ Total number of possible outcomes
P = Number of outcomes in which all four tosses are heads/ Total number of possible outcomes
When a coin is tossed four times, the total number of possible outcomes is 2 × 2 × 2 × 2 = 16.
P = Number of outcomes in which all four tosses are heads/ Total number of possible outcomes
P = 1/16P (you win) = 1/16
The probability of losing is given by the formula
Q = Number of ways to lose/Total number of possible outcomes.
Q = 15/16P (you lose) = 15/16
Average winnings, µ, can be calculated as follows:
Let's say X represents the amount of money you win. When you win, you get $10, and when you lose, you lose $1.
X = -1 when you loseX = 10 when you winUsing the formula
µ = ∑ (X × P), we can calculate the average winnings,
µ = (-1 × 15/16) + (10 × 1/16)µ = -15/16 + 10/16µ = -5/16.
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Given information: A coin is tossed four times. You bet $1 that heads will come up on all four tosses. If this happens, you win $10. Otherwise, you lose your $1 bet.
Therefore, the answers are:
P(you win) = 0.0625
P(you lose) = 0.9375
Average winnings, µ, = -$0.31
Let A be the event of getting heads on the four tosses. Since the coin is tossed four times, the possible outcomes are 2^4 = 16. Thus the probability of getting heads on each toss is 0.5. Therefore, the probability of getting heads on all four tosses is:
P(A) = (0.5)^4
= 0.0625
Let B be the event of not getting heads on all four tosses. Thus:
B = 1 − P(A)
= 1 − 0.0625
= 0.9375
The winning amount for getting all four heads is $10, and the losing amount is $1. Thus, the average winnings is:
µ = (10 × 0.0625) − (1 × 0.9375)
µ = 0.625 − 0.9375
µ = -0.3125
Therefore, the answers are:
P(you win) = 0.0625
P(you lose) = 0.9375
Average winnings, µ, = -$0.31
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A wheel of the given radius is rotating at the indicated rate. radius 9 in., 2100 rpm (a) Find the angular speed (in radians per minute). radians per minute (b) Find the linear speed of a point on the circumference (in ft/min). (Round your answer to the nearest whole number.) ft/min
The linear speed of a point on the circumference is approximately 9895 feet per minute.
(a) To find the angular speed in radians per minute, we need to convert the given rotational speed from rpm (revolutions per minute) to radians per minute. Since there are 2π radians in one revolution, we can use the conversion factor:
Angular speed (in radians per minute) = Rotational speed (in rpm) * 2π
Given that the rotational speed is 2100 rpm, we can calculate the angular speed:
Angular speed = 2100 rpm * 2π ≈ 13194 radians per minute
Therefore, the angular speed of the wheel is approximately 13194 radians per minute.
(b) To find the linear speed of a point on the circumference in feet per minute, we can use the formula:
Linear speed = Angular speed * Radius
Given that the radius of the wheel is 9 inches, we need to convert it to feet:
Radius = 9 inches * (1 foot / 12 inches) = 0.75 feet
Now, we can calculate the linear speed:
Linear speed = 13194 radians per minute * 0.75 feet ≈ 9895 feet per minute
Therefore, the linear speed of a point on the circumference is approximately 9895 feet per minute.
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What is the area of a parallelogram that has a base of 4 ½ inc. and a height of 2 ¼ in. ?
Answer:
4.5
Step-by-step explanation:
What is the quotient of (x3 + 3x2 + 5x + 3) ÷ (x + 1)?
x2 + 4x + 9
x2 + 2x
x2 + 2x + 3
x2 + 3x + 8
Answer:
C
Step-by-step explanation:
C
Suppose a and n are relatively prime such that g.c.da, n=1, prove that \/ b 1 b) If n = 1, we cannot conclude that x=a (mod n) has solutions.
If a and n are relatively prime (gcd(a, n) = 1), it does not guarantee that the equation x ≡ a (mod n) has solutions.
If a and n are relatively prime, denoted by gcd(a, n) = 1, it means that a and n do not have any common factors other than 1. However, this does not guarantee that the equation x ≡ a (mod n) has solutions.
The equation x ≡ a (mod n) represents a congruence relation, where x is congruent to a modulo n. This equation implies that x and a have the same remainder when divided by n.
To have solutions for this congruence equation, it is necessary for a to be congruent to some number modulo n. In other words, a must lie in the residue classes modulo n. However, the fact that gcd(a, n) = 1 does not ensure that a is congruent to any residue modulo n, hence not guaranteeing the existence of solutions for the equation.
Therefore, when n = 1, we cannot conclude that the equation x ≡ a (mod n) has solutions.
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At a candy store, Layla bought 2 pounds of jelly beans and 4 pounds of gummy worms for $46. Meanwhile, Brittany bought 3 pounds of jelly beans and 2 pounds of gummy worms for $41. How much does the candy cost?
X= Y=
Answer:
X = $9.00 for 1 pound of Jelly Beans
Y = $7.00 for 1 pound of Gummy Worms
Step-by-step explanation:
Ok, to start we need to make up some notation. It looks like you've chosen X and Y as your variables which works for me.
First lets say X = cost of one pound of jelly beans and Y = cost of one pound of gummy worms
Now let's put together our equations
2X + 4Y = 46
3X + 2Y = 41
Now we need to solve for one of the variables using substitution or elimination. Since none of the variables are isolated, let's use elimination by multiplying the bottom equation by -2.
2X + 4Y = 46
-6X - 4Y = -82
___________
-4X + 0 = -36
-4X = -36
X = 9 or $9.00 per pound
Now we can plug that back into own of the equations and solve for Y
2(9) + 4Y = 46
18 + 4Y = 46
4Y = 28
Y = 7 or $7.00 per pound
Observa el siguiente polígono y responde la pregunta.
¿Cuál es el perímetro del polígono?
A.
13mn3+4m2+3mn2+2mn
B.
53mn3+4m2+2mn2+2mn
C.
13mn3+3m2+4mn2+3mn
D.
53mn3+3m2+2mn2+3mn
The correct option regarding the perimeter of the polygon in this problem is given as follows:
5/3mn³ + 3mn² + 3m².
What is the perimeter of a polygon?The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.
Hence the expression for the perimeter is given as follows:
1/3mn³ + mn² + m² + 2mn³ + 2mn² + 3m².
Combining the like terms, the perimeter is given as follows:
5/3mn³ + 3mn² + 3m².
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The stock of Company A gained 6% today to $94.87. What was the opening price of the stock in the beginning of the day?
Answer:
Step-by-step explanation:
I don’t know
We want to find the opening price of the stock in the beginning of the day. We will find that the solution is $89.50.
Working with percentages.
Let's say that the opening price of the stock was X.
Then, it is increased 6% to get to $94.87
This means that the 106% of X is equal to $94.87, then we can solve:
1.06*X = $94.87
Where we wrote 106% in decimal form, now we can just solve this for X:
X = $94.87/1.06 = $89.50
This means that the price of the stock in the beginning of the day was $89.50
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There are 2,100 bacteria in a circular petri dish. The dish has a radius of 40 millimeters. What is the approximate population density? (Use 3.14 for π)
Area of a circle = πr2
0.02 bacteria per square millimeter
0.42 bacteria per square millimeter
2.39 bacteria per square millimeter
52.5 bacteria per square millimeter
The approximate Population density in the circular petri dish with a radius of 40 millimeters and 2,100 bacteria is approximately 0.418 bacteria per square millimeter. After rounding up the bacteria becomes 0.42 per sq. meter hence the correct option is b
The approximate population density in the circular petri dish, we need to calculate the area of the dish and divide the total number of bacteria by that area.
The formula to calculate the area of a circle is: A = πr^2, where A represents the area and r is the radius.
In this case, the radius of the petri dish is given as 40 millimeters. Therefore, the area can be calculated as follows:
A = 3.14 * (40)^2
A = 3.14 * 1600
A ≈ 5024 square millimeters
Next, we divide the total number of bacteria (2,100) by the calculated area to find the population density:
Population density = 2,100 bacteria / 5024 square millimeters
Population density ≈ 0.418 bacteria per square millimeter
Therefore, the approximate population density in the circular petri dish is 0.418 bacteria per square millimeter.
in scientific calculations, it is customary to use the value of π as 3.14 for simplicity, even though it is an irrational number with a longer decimal representation.
When writing about scientific concepts or calculations, it is crucial to ensure academic integrity by avoiding plagiarism. This can be done by understanding and explaining the concepts in your own words, citing any external sources used, and providing accurate calculations based on the given information.
In conclusion, the approximate population density in the circular petri dish with a radius of 40 millimeters and 2,100 bacteria is approximately 0.418 bacteria per square millimeter. After rounding up the bacteria becomes 0.42 per sq. meter hence the correct option is b
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You have 1/8 gallon of melted crayon wax. You pour the wax equally into 8 different molds to make new crayons. What fraction of a cup of melted wax is in each mold? Think: 1 gallon is 16 cups.
PLS HELP
Answer:
2 cups
Step-by-step explanation:
16/8=2 so it is 2 cups
Please help me with this, please
Answer:
Hopefully it makes sense
Step-by-step explanation:
Good luck
1. The two sample t-test The carapace lengths (measure in mm) of crawfish (Palinurus vulgaris) captured in streams in Devon and Cornwall were measured. The data is given below: Carapace Length (in mm) Devon: 170,111,135,182,121,174,169,133,141,147,159,163 Cornwall: 146, 97, 102, 181, 107, 118,131,155,127,130, 129 a. Do you have reason to believe the two populations of crawfish do both have the same mean carapace length? Use the t test. b. Can you answer the question in a. using the Wilcoxon rank sum test? (See note below.) c. Compare the results obtained in a. and b. Are you surprised? The Wilcoxon Rank Sum Test - The Wilcoxon Rank Sum (WRS) test is the distribution free alternative to the t-test. It does not consider the actual value of the observations but only their relative position in the combined set of observations from the two samples, A and B. To use the WRS, you combine the observations of the two samples, order them from smallest to the largest, given rank 1 to the smallest observation (in either sample), rank 2 to the second smallest, and so on, giving rank n+m to the largest observation. The null hypothesis is that the two distributions are the same. The test statistic, WA or WB is the sum of the ranks of the observations in one of the samples. Reference Distribution For samples of similar sized with a combined number of observation in excess of 20, if H0 is true WA will have a distribution that is approximately Normal with µ = (nA)(nA+nB+1) 2 and variance σ 2 A = nAnB(nA+nB+1) 12 . Note that the variance is the
a. To determine if the two populations of crawfish from Devon and Cornwall have the same mean carapace length, we can use the two-sample t-test.
This test compares the means of two independent samples to assess whether they are significantly different.
We can calculate the sample means and sample standard deviations for both groups:
Next, we calculate the t-value using the formula:
To determine if this t-value is statistically significant, we need to compare it to the critical value from the t-distribution for the given degrees of freedom
If the calculated t-value falls outside the critical region, we can reject the null hypothesis and conclude that the two populations have different mean carapace lengths. Otherwise, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in the mean carapace length between the two populations.
b. To answer the question using the Wilcoxon rank sum test, we need to combine the observations from both samples, assign ranks based on their relative positions, and calculate the sum of ranks for one of the samples (either Devon or Cornwall). The null hypothesis for this test is that the two distributions are the same.
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If the population density for the city is 10,000 people per square mile, what is the population of the city?
Answer:
10,000 * (square miles in the city)
Step-by-step explanation:
You've been given a population of people for every square mile (10,000) and you're trying to figure out how many people live in the city.
To convert people per a square mile into people, you'll need to know how many square miles you have. I assume you've been given this as the size of the city or how many square miles the city takes up. Just multiply the amount of people per a square mile by the amount of square miles in the city, and it'll give you the population of people in every square mile of the city (the population).
Note:
This is people per a square mile, or in other words, for every square mile of city, 10,000 people live there. If you have 2 square miles of city, then 2 * 10,000 = 20,000 people live there.
the probability that an at home pregnancy test will correctly identify a pregnancy is 0.98. suppose 11 randomly selected pregnant women with typical hormone levels are each given the test. rounding your answer to four decimal places, find the probability that all 11 tests will be positive at least one test will be negative
The probability that all 11 tests will be positive is 0.8007
The probability that at least one test will be negative is 0.1993
Finding the probability that all 11 tests will be positiveFrom the question, we have the following parameters that can be used in our computation:
p = 0.98
n = 11
The probability that all 11 tests will be positive is
P = pⁿ
So, we have
P = 0.98¹¹
Evaluate
P = 0.8007
Finding the probability that at least one test will be negativeHere, we use
P = 1 - P(No negative)
So, we have
P = 1 - 0.8007
Evaluate
P = 0.1993
Hence, the probability is 0.1993
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Write a function for the graph.
Answer:
(x-4)^2 -4 = y
Step-by-step explanation:
The graph is translated 4 to the right and 4 downwards.
Use induction to prove that if A1, A2,...,An and B are sets,
then (A1 −B) ∩(A2 −B) ∩...∩(An−B) = (A1 ∩A2 ∩...∩An) −B.
It is proved that if A1, A2,...,An and B are sets, then (A1 −B) ∩(A2 −B) ∩...∩(An−B) = (A1 ∩A2 ∩...∩An) −B.
To prove the given statement using induction, we need to show that it holds true for a base case and then demonstrate that if it holds for an arbitrary value of 'n', it also holds for 'n + 1'.
Base Case (n = 1):
Let's consider the base case where 'n = 1'. We have two sets A1 and B, and we want to prove that (A1 - B) = (A1 - B).
(A1 - B) ∩ (A1 - B) = (A1 - B) [Using the definition of set difference]
This satisfies the given statement for the base case.
Inductive Step
Assuming that the given statement holds for 'n = k', let's prove that it holds for 'n = k + 1'.
We have sets A1, A2, ..., Ak+1, and B. We want to prove that:
(A1 - B) ∩ (A2 - B) ∩ ... ∩ (Ak+1 - B) = (A1 ∩ A2 ∩ ... ∩ Ak+1) - B
Using the inductive hypothesis, we can rewrite the left-hand side (LHS) as:
[(A1 ∩ A2 ∩ ... ∩ Ak) - B] ∩ (Ak+1 - B)
To prove the equality, we need to show that each side is a subset of the other.
1. LHS ⊆ RHS:
Let x be an arbitrary element in LHS. This means that x belongs to each set in the intersection on the LHS: (A1 - B), (A2 - B), ..., (Ak+1 - B). By the definition of set intersection, x also belongs to (A1 - B), (A2 - B), ..., (Ak - B), and (Ak+1 - B).
Since x belongs to each set in the intersection (A1 - B) ∩ (A2 - B) ∩ ... ∩ (Ak+1 - B), it follows that x belongs to (A1 ∩ A2 ∩ ... ∩ Ak) - B. Therefore, x ∈ RHS.
2. RHS ⊆ LHS:
Now let y be an arbitrary element in RHS. This means that y belongs to (A1 ∩ A2 ∩ ... ∩ Ak+1) - B. By the definition of set difference, y belongs to (A1 ∩ A2 ∩ ... ∩ Ak+1) and y does not belong to B.
Since y belongs to (A1 ∩ A2 ∩ ... ∩ Ak+1), it implies that y belongs to each set A1, A2, ..., Ak+1. By the definition of set difference, y does not belong to B, so y also belongs to each set (A1 - B), (A2 - B), ..., (Ak+1 - B).
Therefore, y ∈ LHS.
Since we have shown that LHS ⊆ RHS and RHS ⊆ LHS, it follows that (A1 - B) ∩ (A2 - B) ∩ ... ∩ (Ak+1 - B) = (A1 ∩ A2 ∩ ... ∩ Ak+1) - B.
By the principle of mathematical induction, the statement holds for all 'n', completing the proof.
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