The statement is true.
To prove this, we will use a proof by contradiction.
Assume that all of the sets {ai lie i} are finite. Then, for each set ai, there exists a finite number of elements in that set. Therefore, the union of all of these sets will also be finite.
However, we are given that the union of all the sets is countably infinite. This means that there exists a countable list of elements in the union.
Let's construct this list:
- First, list all of the elements in a1.
- Then, list all of the elements in a2 that are not already in the list.
- Continue this process for all of the remaining sets.
Since the union is countably infinite, this process will never terminate and we will always have elements to add to our list.
But this contradicts the fact that each set is finite. If each set has a finite number of elements, then there can only be a finite number of unique elements in the union.
Therefore, our assumption that all of the sets are finite must be false. At least one of the sets must be countably infinite.
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Olga has a drawer of socks with four red pears, two gray pears, and a black hairs. she also has a bag of hair bands with 10 red, six gray, and 24 black hair bands if she selects randomly what is the probability that olga will select a red pair of socks and a red hairband
The probability that olga will select a red pair of socks and a red hairband is 14/47
Probability defines the likelihood of an event occurring. There are many situations in real life where we may need to predict the outcome of an event. We may or may not be certain of the outcome of the event. In such cases, we say that there is a probability that the event will or will not occur. Probability in general has great applications in games, in business for probabilistic predictions, and also probability has extensive applications in this new field of artificial intelligence.
The probability of an event can be calculated using the probability formula simply by dividing the number of favorable outcomes by the total number of possible outcomes. The value of the probability of an event occurring can lie between 0 and 1 because the number of favorable outcomes can never exceed the total number of outcomes. Also, a favorable number of results cannot be negative
For an experiment that has "n" number of outcomes, the number of favorable outcomes can be denoted by x. The formula for calculating the probability of an event is as follows.
Probability (event) = favorable outcomes/total outcomes = x/n
Olga has a drawer of socks with four red pears, two gray pears, and a black hairs. she also has a bag of hair bands with 10 red, six gray, and 24 black hair bands
Therefore total number of items are = 4 + 2 + 1 + 10 + 6 + 24 = 47
Probability that olga will select a red pair of socks and a red hairband is 4/47 + 10/47
= 14/47
Hence the probability that olga will select a red pair of socks and a red hairband is 14/47
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The function f(x) = 200 × (1.098)x represents a village's population while it is growing at the rate of 9.8% per year.
Create a table to show the village's population at 0, 2, 4, 6, 8, and 10 years from now.
Use your table to create a graph that represents the village's population growth.
When the population doubles from its current size, the village will need to dig a new water well. To the nearest half of a year, about how long before it is time for the village to dig the new well?
The time for the village to dig the new well is about 7 1/2 years. At this time the population doubles from its current size.
How to graph a function?For a function f(x) = y, the steps to draw a graph for the given function are as follows:
Consider certain values of x Substitute x values in the given function to obtain y valuesPlot the x and y values in the graphJoin the points to know the variation.Calculation:It is given that,
f(x) = 200 × [tex](1.098)^x[/tex]
Village's population growing rate = 9.8%
Creating a table for x and f(x) values:
x: 0 2 4 6 8 10
f(x = 0) = f(0) = 200 × [tex](1.098)^0[/tex] = 200
f(x = 2) = f(2) = 200 × [tex](1.098)^2[/tex] = 241
f(x = 4) = f(4) = 200 × [tex](1.098)^4[/tex] = 291
f(x = 6) = f(6) = 200 × [tex](1.098)^6[/tex] = 350
f(x = 8) = f(8) = 200 × [tex](1.098)^8[/tex] = 423
f(x = 10) = f(10) = 200 × [tex](1.098)^1^0[/tex] = 509
So,
(x, f(x)): (0, 200) (2, 241) (4, 291) (6, 350) (8, 423) (10, 509)
Plotting these points in the graph and joining the points, the graph is shown below.
Finding the time for the village to dig a new well:
It is given that the population doubled from its current size, the village will need to dig new water well.
The current size of the village = 200
If it doubles for a certain time 't' = 400
So,
400 = 200 × [tex](1.098)^t[/tex]
⇒ [tex](1.098)^t[/tex] = 2
⇒ log[tex](1.098)^t[/tex] = log 2
⇒ t log(1.098) = 0.301
⇒ t × 0.04 = 0.301
∴ t = 7.525 years
Therefore, the village's population becomes double at the time 7 and half years from now. So, after 7 1/2 years, the village needs to dig a new well.
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URGENT!!! Write a compound interest function to model the following situation. Then, find the balance after the given number of years.
Answer:
The answer to our question is A = 16,100(1.001)12t; $17,510
Step-by-step explanation:
I hope this helps and have a good day!
the value of dimes in a vending machine is $1.70 more than what it contains in quarters.if there are 199 coins in all how many dimes and quarters are there
Using a system of equations, it is found that there are 147 dimes and 52 quarters in the machine.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the variables are:
Variable x: Number of dimes.Variable y: Number of quarters.There are 199 coins, hence:
x + y = 199 -> y = 199 - x.
The value of dimes(each is worth $0.1) is $1.7 more than the sum of the quarters(each is worth $0.25), hence:
0.1x - 0.25y = 1.7.
Since y = 199 - x:
0.1x - 0.25(199 - x) = 1.7.
0.35x = 51.45
x = 51.45/0.35
x = 147
y = 199 - 147 = 52
There are 147 dimes and 52 quarters in the machine.
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Sean tried to drink as fast as he could. He drank the slushy at a constant rate. There were originally 275 milliliters of slushy in the cup. After 13 seconds, 210 milliliters of slushy remained. (Please help and explain.)
How fast did Sean drink?
How long did it take Sean to drink all the slushy?
a. Sean drank at a rate of 5 milliliters per second.
b. It took Sean 55 s to drink all the slushy.
a. How to find the rate at which Sean drank?Since Sean darnak at a constant rate and originally there were 275 milliliters of slushy in the cup. After 13 seconds, 210 milliliters of slushy remained.
The rate at which Sean drank, r = Change in volume of slushy/time for change
= (275 mL - 210 mL)/13 s
= 65 mL/13 s
= 5 mL/s
So, Sean drank at a rate of 5 milliliters per second.
b. How long did it take Sean to drink all the slushy?The time it takes Sean to drink all the slushy is given by
T = V/r where
V = total volume of slushy = 275 milliliters and r = rate of drinking by Sean = 5 mL/sSo, T = V/r
= 275 mL/5 mL/s
= 55 s
So, it took Sean 55 s to drink all the slushy.
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Each element in a population has an equal chance of being chosen for inclusion in the sample independent of any other event in the process. This is known as _________.
Each element in a population has an equal chance of being chosen is Random selection.
Given that each element in a population has an equal chance of being chosen for inclusion in the sample independent of any other event in the process.
Sampling is an umbrella term for all methods used to take samples from populations. The main purpose of sampling is to ensure that a chosen sample reflects the desired characteristics of a population so that a statistical study can produce accurate and reasonable results. Sampling techniques include random sampling, stratified sampling and many others.
As we know, random selection is the process of selecting a small group of people from a larger group to participate in a study. Each person has an equal chance of being selected, giving each person in the group an equal chance to participate.
Hence, Each element in a population has an equal chance of being chosen for inclusion in the sample independent of any other event in the process is known as random selection.
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5. A fair coin is flipped 10 times. (a) What is the probability that there are an equal number of heads and tails
Answer:
you have a 50/50 chance to get equal number of heads & tails
let n be a natural number such that GCF(n, 70)= 10, and LCM(n, 70)=210. Find N. (Hint: Do you remember that LCM(A,B) x GCF(A, B)= A x B?)
The natural number , n is 30
How to determine the value
Given;
GCF(n, 70) = 10
LCM(n, 70) = 210
We have that;
LCM(A,B) x GCF(A, B)= A x B
Then,
(n, 70) = n × 70 = 70n
10 × 210 = 2100
Equate the two expressions
70n = 2100
n = [tex]\frac{2100}{70}[/tex]
n = [tex]30[/tex]
Thus, the natural number , n is 30
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If a truck traveled 75 miles in 3 hours, how many miles would the truck have traveled per minute?
The truck travelled 0.42 miles per minute
How to determine the miles per minute?The given parameters are:
Distance = 75 miles
Time = 3 hours
Convert time to minutes
Time = 3 * 60 minutes
Time = 180 minutes
The speed is calculated as:
Speed = Distance/Time
This gives
Speed = 75 miles/180 minutes
Evaluate
Speed = 0.42 miles per minutes
Hence, the truck travelled 0.42 miles per minute
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Which choices are equivalent to the quotient below? Check all that apply. √16 √8
The equivalent quotient of the expression √16/√8 are: √2 and √4 / √2
How to find the equivalent of a quotient?We want to find the equivalent quotient of the given expression; √16 / √8
Now, we know that the square root of 16 is equal to 4 i.e. √16 = 4
Thus, applying that to our expression gives;
√16/√8 = 4/√8
Now, square root of 8 which is √8 can also be expressed as; 2√2
Thus;
4/√8 = 4/2√2 = 2/√2
The steps to rationalize the denominator, to get rid of all radicals that are in the denominator are as follows;
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.
Step 2: Make sure all radicals are simplified.
Step 3: Simplify the fraction if needed.
If we rationalize the denominator of our expression, we have;
2/√2 × √2/√2 = 2√2/ 2 = √2
Also, 2/√2 can be expressed as √4 /√2
Therefore, The equivalent quotient of the expression √16/√8 are: √2 and √4 / √2
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Someone help me please
Answer: 12
Step-by-step explanation:
Let's say that the length needed to be found is x
As the two lengths of both triangles are parallel to one another, we can use the equation to find x
6/x = (6+4)/20
6/x = 1/2
x = 12
what is the mode of 250 100 75 200 150 100?
[tex]\textbf{Heya !}[/tex]
the mode is the number that occurs the most
All numbers here occur once, except [tex]\sf{100}[/tex].
so 100 is the mode.
`hope it was helpful to u ~
D is the centroid of triangle abc. what is the value of x when bd=2x+40 and bf=18x
The value of x= 20 units
BD = BF
BD =2x + 40
BF = 18x
Equate the expressions thus:
2x + 40 = 2/3(18x)
2x + 40 = 12x
Collect like terms
2x - 12x = - 40
-2x = - 40
Divide both sides by - 2
x = 20
Hence, x = 20 units
The centroid is the center point of the item. The factor in which the 3 medians of the triangle intersect is known as the centroid of a triangle. it is also defined because of the point of intersection of all the 3 medians. The median is a line that joins the midpoint of a facet and the opposite vertex of the triangle.
Then, we can calculate the centroid of the triangle by means of taking the common of the x coordinates and the y coordinates of all of the three vertices. So, the centroid method can be mathematically expressed as G(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/three).
The centroid of a triangle is the point at which the 3 medians coincide. The centroid theorem states that the centroid is 23 of the space from each vertex to the midpoint of the opposite aspect.
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A couple quick algebra 1 questions for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
By the way the 2nd and 3rd attachments (or the ones with the chart filled out already) are my first 2 attempts, but I got both of them mostly all wrong. So please use those as what not to do lol. And the first attachment (without the chart filled out) is the actual question and the one I need to get answers for, tysm!
All the relevant transformations are as shown in the explanations below.
How to Interpret Transformations?We are given the original function as y = x and told the transformation is as follows;
1) y = x + 5; This means that the function was shifted 5 units to the left.
2) y = ³/₈x; This means that the function was vertically stretched by a scale factor of ³/₈.
3) y = -x - 2 or y = -(x + 2); This means that the function was first shifted by 2 units to the left before being reflected over the x-axis.
4) y = 6x + 1; This means that the function was stretched by a factor of 6 and then shifted 1 unit to the left.
5) y = x - 11; This means that the function was shifted 11 units to the right.
6) y = 8x; This means that the function was vertically stretched by a scale factor of 8.
7) y = -1/3x; This means that the function was horizontally stretched by a scale factor of 1/3.
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Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit
Answer:
106
Step-by-step explanation:
If 1 student gets a 1, and 106 get a 100, then the mean will be 99.99, and 106 students will get the extra credit.
Describe a relationship that can be modeled by the function represented by the graph, and explain how the function models the relationship. Identify and interpret the key features of the function in the context of the situation you described in part A.
The graph is a decreasing exponential graph.
How to illustrate the information?Part A: The given graph is decreasing the exponential graph since the graph is decreasing rapidly as the value of x is increasing the value of y is decreasing very rapidly.
Part B: The graph shown here is maybe the case of any electric appliance which is of cheap quality. The time when it is brought then its performance and its life is awesome after that its life is decreasing rapidly.
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the ratio of peters money to henrys money is 4 : 3 there at first. after perter spent 12 dollars they had an equal amount of money each. How much money id peter have at first
Using a system of equations, it is found that Peter had $48 at first.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the variables are:
Variable x: Peter's money.Variable y: Henry's money.The ratio of peters money to henrys money is 4 : 3, hence:
[tex]\frac{x}{y} = \frac{4}{3}[/tex]
After Peter spent $12, they had the same amount, hence:
y = x - 12.
Then, replacing in the ratio:
[tex]\frac{x}{y} = \frac{4}{3}[/tex]
[tex]\frac{x}{x - 12} = \frac{4}{3}[/tex]
4(x - 12) = 3x
x = 48.
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Match each polynomial with its correct, most specific classification.
3x^2yz
2x^2+3xy-6y^2
3pq^2-3p
2xy+3x^2-2y^2-4x^2
The polynomials are classified as follows:
3x²yz: Monomial.2x² + 3xy - 6y²: Trinomial.3pq² - 3p: Binomial.2xy + 3x² - 2y² - 4x²: Polynomial.How are the polynomials classified?Depends on the number of terms, as follows:
3x²yz: Monomial, as it has one term.2x² + 3xy - 6y²: Trinomial, as it has three terms.3pq² - 3p: Binomial, as it has two terms.2xy + 3x² - 2y² - 4x²: Polynomial, as it has four terms.More can be learned about polynomials at https://brainly.com/question/24662212
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The price of the product decreased by 10%and then decreased by 20% . calculate total percentage change between the final and initial prices
What is percentage change?
By quantifying the difference between two numbers and expressing it as an increase or decrease, the Percentage Change Calculator (percent change calculator) may determine the percentage change.
The initial price [tex]V_{1}[/tex] (100%) was decreased by 10%., then this reduced price is reduced more by 20% that becomes the final price [tex]V_{2}[/tex] which is 72.
Calculate percentage change from [tex]$V_{1}=100$[/tex] to [tex]$V_{2}=72$[/tex].
=[tex]& \frac{\left(V_{2}-V_{1}\right)}{\left|V_{1}\right|} \times 100 \\[/tex]
[tex]=& \frac{(72-100)}{|100|} \times 100 \\[/tex]
[tex]=& \frac{-28}{100} \times 100 \\[/tex]
[tex]=&-0.28 \times 100 \\[/tex]
[tex]=&-28 \% \text { change } \\[/tex]
[tex]=& 28 \% \text { decrease }[/tex]
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What is the value of the expression below when w = 9 and x = 5?
10w + 4x
Answer:
110
Step-by-step explanation:
10w + 4x
10 (9) + 4 (5)
90 + 20
110
The answer is 110.
Hope it helps!
Have a nice day:)
Mia invests $5000 in an index fund after 1 year the funds share price has increased by 8% how much did mia gain
A year later, Mia gained $400.
Calculation of interestMia invests in an index fund= $5000
The share price of the fund has grown by 8% after a year.
So, one year later Mia's investment would also increase by 8% since her invested share value has grown by 8%.
Hence, the amount of share would be = ($5000 + 8% of $5000) =($5000+$5000*8/100) = ($5000+$400) = $ 5400
Therefore, Mia's gain is $400.
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In a year of 365 days Paul skipped school for the months of January and February.
He had summer vacations for the months June to August. How many days of the
year (excluding Sundays) did he attend school?
The days of the year (excluding Sundays) paul attends school is 183 days
CalendarTotal days in a year = 365
Months Paul skipped school = January, February, June, July August
= 31 + 28 + 30 + 31 + 31
= 151 days
Days he attend school excluding Sunday = Total days in a year - (Months Paul skipped school + Sundays)
= 365 - (151 + 31)
= 365 - 182
= 183 days
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Last week, Layla's Ice Cream Shoppe sold 11 sundaes with nuts and 39 without. what is the ratio of the number of sundaes without nuts
The ratio of the number of sundaes with nuts to the number of sundaes without nuts is; 11:39
How to find the sharing ratio?We are given;
Number of sundaes with nuts sold = 11
Number of sundaes without nuts sold = 39
Thus, the ratio of the number of sundaes with nuts to the number of sundaes without nuts is;
11:39
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Can someone help please
Answer:
x=7 y=3
Step-by-step explanation:
-x+6y=11
2x-3y=5
-2x+12y=22
2x-3y=5
Add both equations
-2x+2x-3y+12=22+5
9y=27
y=3
plug y back in
-x+18=11
-x=-7
x=7
I SHOULD ALREADY KNOW THIS
[tex]t \geqslant 35[/tex]
Since there are at least 35 tic tacs, there has to be 35 tic tacs or more in the box.what is the height of figure 1?
Answer:
4
Step-by-step explanation:
If the rr is equal to 25%, what is the value of the money multiplier (simple deposit multiplier)? if $35,000 in cash is taken from under your mattress and deposited into a bank, predict the impact on the money supply in the economy.
The money supply in the economy is $1025302.6
Bank loans or invests its excess reserves to earn greater hobby. A one-dollar boom inside the financial base reasons the money deliver to boom through more than one dollar. The growth inside the money supply is the cash multiplier.
The money multiplier is the primary can used to calculate what a change in reserves could do to the money supply. The method for the cash multiplier is 1/r where r is the reserve ratio. once one has calculated the money multiplier, they might then multiply that through the exchange in reserves.
The money multiplier is critical in macroeconomics because it determines the cash supply, which affects interest charges. it is also critical in banking because it impacts economic coverage and the steadiness of the banking region.
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There are 15 unmatched socks in your drawer. 9 are black, 5 are white, and 1 is brown. What is the probability that if you already randomly choose one of the black socks, that you will choose another black sock?
Answer: 4/7
Step-by-step explanation:
You pick one black sock, meaning there are 14 socks remaining in the drawer, 8 of which are black. Thus, the probability is 8/14, which simplifies to 4/7.
Find the surface.area of the composite.figure...
11 in.
3 in.
6 in.
3 in.
H
8 in.
SA =
3 in.
6 in.
11 in.
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Enter
Solve 3[-x + (2 x +1)]=x-1
Question :
Solve 3 [-x + (2x + 1) ] = x - 1Solution :
⇒ 3 [-x + 2x + 1] = x - 1
⇒ 3 [x + 1] = x - 1
⇒ 3 × [x + 1] = x - 1
⇒ 3x + 3 = x - 1
⇒ 3x - x + 3 = -1
⇒ 2x + 3 = -1
⇒ 2x = -1 - 3
⇒ 2x = -4
⇒ x = -4 / 2
⇒ x = -2
Simplifying
3[-1x + (2x + 1)] = x + -1
Reorder the terms:
3[-1x + (1 + 2x)] = x + -1Remove parentheses around (1 + 2x)
3[-1x + 1 + 2x] = x + -1Reorder the terms:
3[1 + -1x + 2x] = x + -1Combine like terms: -1x + 2x = 1x
3[1 + 1x] = x + -1[1 * 3 + 1x * 3] = x + -1[3 + 3x] = x + -1Reorder the terms:
3 + 3x = -1 + xSolving
3 + 3x = -1 + xSolving for the variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-1x' to each side of the equation.
3 + 3x + -1x = -1 + x + -1xCombine like terms: 3x + -1x = 2x
3 + 2x = -1 + x + -1xCombine like terms: x + -1x = 0
3 + 2x = -1 + 03 + 2x = -1Add '-3' to each side of the equation.
3 + -3 + 2x = -1 + -3
Combine like terms: 3 + -3 = 0
0 + 2x = -1 + -32x = -1 + -3Combine like terms: -1 + -3 = -4
2x = -4Divide each side by '2'.
x = -2Simplifying
x = -2