a) f is onto because for every element c in set C, there exists an element b in set B such that f(b) = c.
b) g is one-to-one because if g(a1) = g(a2), then a1 = a2.
c) if fog is a bijection, then g is onto if and only if f is one-to-one.
d) fog is a bijection, but g is not onto and f is not one-to-one.
a) To prove that if fog is onto, then f must be onto, we need to show that for every element c in set C, there exists an element a in set A such that f(a) = c.
Given that fog is onto, it means that for every element c in set C, there exists an element a in set A such that fog(a) = c. Since fog(a) = f(g(a)), this implies that for every element c in set C, there exists an element b = g(a) in set B such that f(b) = c.
Therefore, f is onto because for every element c in set C, there exists an element b in set B such that f(b) = c.
b) To prove that if fog is one-to-one, then g must be one-to-one, we need to show that if fog(a1) = fog(a2), then a1 = a2.
Assume that fog is one-to-one, so if fog(a1) = fog(a2), then it implies that a1 = a2. Since fog(a1) = f(g(a1)) and fog(a2) = f(g(a2)), if f(g(a1)) = f(g(a2)), it follows that g(a1) = g(a2) because f is a function.
Therefore, g is one-to-one because if g(a1) = g(a2), then a1 = a2.
c) To prove that if fog is a bijection, then g is onto if and only if f is one-to-one, we need to prove both directions:
(i) If fog is a bijection, and g is onto, then f is one-to-one.
Assume that fog is a bijection, which means it is both one-to-one and onto. If g is onto, it implies that for every element b in set B, there exists an element a in set A such that g(a) = b. Since fog is one-to-one, it implies that for every element a1 and a2 in set A, if fog(a1) = fog(a2), then a1 = a2. Now, let's assume that f is not one-to-one, which means there exist elements b1 and b2 in set B such that f(b1) = f(b2), but b1 ≠ b2. Since g is onto, there exist elements a1 and a2 in set A such that g(a1) = b1 and g(a2) = b2. This means that fog(a1) = f(g(a1)) = f(b1) = f(b2) = f(g(a2)) = fog(a2), but a1 ≠ a2, which contradicts fog being one-to-one. Therefore, f must be one-to-one.
(ii) If fog is a bijection, and f is one-to-one, then g is onto.
Assume that fog is a bijection, which means it is both one-to-one and onto. Also, assume that f is one-to-one. We want to prove that g is onto. Let b be an element in set B. Since fog is onto, there exists an element a in set A such that fog(a) = f(g(a)) = b. Since f is one-to-one, there can only be one element a that maps to b. Therefore, g(a) must equal b. Hence, for every element b in set B, there exists an element a in set A such that g(a) = b, indicating that g is onto.
Therefore, if fog is a bijection, then g is onto if and only if f is one-to-one.
d) Examples of functions f and g such that fog is a bijection, but g is not onto and f is not one-to-one:
Let A = {1, 2} (two elements), B = {3} (one element), and C = {4, 5} (two elements).
Define function g: A → B as g(1) = g(2) = 3 (constant mapping).
Define function f: B → C as f(3) = 4.
Then, the composition fog: A → C is fog(1) = fog(2) = f(g(1)) = f(g(2)) = f(3) = 4.
In this example, fog is a bijection because it is both one-to-one and onto. However, g is not onto because B contains only one element. Also, f is not one-to-one because f(3) = 4, and there is no restriction on the pre-image of 4 (both elements in A map to 3).
Therefore, fog is a bijection, but g is not onto and f is not one-to-one.
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The sum of two numbers is 48. The second number is twice the first
Write and solve a system of equations to find the numbers.
Answer:
The equations are
X + Y = 48 ----Eq (1)
Y = 2 X -----Eq (2)
Solution of the equations and the two number are
X = 16, Y = 32
Step-by-step explanation:
Let the two numbers be X and Y
Given -
The sum of two numbers is 48.
Thus, X + Y = 48 ----Eq (1)
The second number is twice the first
Thus, Y = 2 X -----Eq (2)
Substituting the value of Y in equation (1), we get -
X + 2X = 48
3 X = 48
X = 16
Y = 2 X = 32
PLZZ ILL GIVE BRAINLIESTTT
Answer:
give me poi tasjnsaklaalwkdbfbdjwo
A long straight wire carrying a 4-A current is placed along the x-axis as shown in the figure. What is the direction of the magnetic field at a point P due to this wire? Q Tap image to zoom 0 along the +x-axis 0 out of the plane of the page 0 along the -x-axis 0 into the plne of the page into the plane of the page 0 along the ty-axis
The direction of the magnetic field at P is perpendicular to both the current direction and the direction of the curl of our fingers.
The direction of the magnetic field at point P due to the current-carrying wire can be determined using the right-hand rule. If we point our right thumb in the direction of the current (which is along the positive x-axis), and curl our fingers toward the point P, then the direction of the magnetic field at P is perpendicular to both the current direction and the direction of the curl of our fingers.
In this case, since the wire is straight and lies in the x-y plane, the direction of the magnetic field at point P will be perpendicular to the plane of the page, and will either be pointing into or out of the page. To determine which direction, we need to know the orientation of point P relative to the wire. If point P is above the wire, then the magnetic field will be pointing into the page, and if it is below the wire, then the magnetic field will be pointing out of the page.
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Help me please I will give points
NO FAKE ANSWERS
Answer: 2,3,4
Step-by-step explanation:
Apple’s is represented by 8 and oranges is represented by 5. So the ratio 8:5 means every 8 apples their are 5 oranges
Chad earns $4 each day walking his neighbor's dog. He spends $8 purchasing dog treats
for the dog. Owen spends $3 each day at the local coffee shop. He has $13 saved from a
birthday gift. How many days until the boys have an equal amount of money?
Find with proof the sum from i = 1 to n of 2^i for each n >= 1. Find with proof the sum from i = 1 to n of 1/(i(i+1)) for each n >= 1. Prove that n! > 2^n for each n >= 4.
Prove sqrt(2) is irrational.
Find with proof the sum of the first n odd positive integers.
If A is the set of positive multiples of 8 less than 100000 and B is the set of positive multiples of 125 less than 100000, find |A intersect B|.
Find |A union B|.
There are 7 students on math team, 3 students on both math and CS team, and 10 students on math team or CS team. How many students on CS team?
The sum from i = 1 to n of 2^i is 2(2^n - 1), the sum from i = 1 to n of 1/(i(i+1)) is n/(n+1), n! > 2^n for n ≥ 4, and therefore, sqrt(2) is irrational. The intersection of sets A and B has |A ∩ B| elements, the union of sets A and B has |A ∪ B| elements, and the number of students on the CS team is 6.
Let's break down the questions and provide the proofs and solutions step by step:
Sum of powers of 2: We want to find the sum from i = 1 to n of 2^i for each n ≥ 1. We can use the formula for the sum of a geometric series to simplify the expression:
The sum of a geometric series is given by the formula Sn = a(r^n - 1)/(r - 1), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 2, r = 2, and we need to find Sn.
Plugging in the values, we get Sn = 2(2^n - 1)/(2 - 1) = 2(2^n - 1).
Therefore, the sum from i = 1 to n of 2^i is 2(2^n - 1).
Sum of fractions: We want to find the sum from i = 1 to n of 1/(i(i+1)) for each n ≥ 1. We can rewrite the expression as follows:
1/(i(i+1)) = 1/i - 1/(i+1).
Now, we can observe that the terms cancel out in pairs when we sum them. The first term 1/1 remains, and the last term 1/(n+1) remains as well.
Therefore, the sum from i = 1 to n of 1/(i(i+1)) is 1 - 1/(n+1) = n/(n+1).
Proof of n! > 2^n: We will prove this by induction. The base case is n = 4: 4! = 24 > 2^4 = 16.
Now, assume the inequality holds for some k ≥ 4, i.e., k! > 2^k.
We need to prove it for k + 1: (k + 1)! = (k + 1) * k! > (k + 1) * 2^k (since k! > 2^k by the induction hypothesis).
It suffices to show that (k + 1) * 2^k > 2^(k + 1), which simplifies to k + 1 > 2.
Since k ≥ 4, the inequality holds.
Therefore, by induction, we can conclude that n! > 2^n for each n ≥ 4.
Proof that sqrt(2) is irrational: We will prove this by contradiction. Assume that sqrt(2) is rational, i.e., sqrt(2) can be expressed as a ratio of two integers p and q in its simplest form, where q ≠ 0.
sqrt(2) = p/q.
Squaring both sides, we get 2 = p^2/q^2.
Rearranging, we have p^2 = 2q^2.
This implies that p^2 is even, and thus p must be even.
Let p = 2k, where k is an integer.
Substituting back, we have (2k)^2 = 2q^2, which simplifies to 4k^2 = 2q^2.
Dividing by 2, we get 2k^2 = q^2.
This implies that q^2 is even, and thus q must be even.
However, if both p and q are even, then p/q is not in its simplest form, contradicting our initial assumption.
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Diana looks up at an angle of 57 and sees a hot air balloon 150 meters away. To the nearest meter, what is the value of x,
the height of the hot air balloon above Diana's head? (Show work on paper)
Answer:
230.98 meters
Step-by-step explanation:
Answer:
126m
Step-by-step explanation:
Use sine because we want to find the opposite length but only know the hypotenuse.
evaluate the line integral, where c is the given curve. ∫c xy⁴ ds, c is the right half of the circle x² + y² = 9 oriented counterclockwise
To evaluate the line integral ∫c xy⁴ ds, we need to parameterize the curve c, then substitute into the integrand, and integrate with respect to the parameter.
The right half of the circle x² + y² = 9 can be parameterized as x(t) = 3cos(t), y(t) = 3sin(t) for t in [0, pi]. Note that this parameterization traces out the right half of the circle oriented counterclockwise.
Now, we can express ds as ds = sqrt(dx/dt² + dy/dt²) dt. Using the parameterization x(t) = 3cos(t), y(t) = 3sin(t), we get dx/dt = -3sin(t) and dy/dt = 3cos(t). Thus, ds = sqrt((-3sin(t))² + (3cos(t))²) dt = 3dt.
Substituting x(t) = 3cos(t), y(t) = 3sin(t), and ds = 3dt into the integrand xy⁴, we get xy⁴ = (3cos(t))(3sin(t))⁴ = 81/4 sin⁴(t) cos(t).
So, the line integral becomes:
∫c xy⁴ ds = ∫₀ᴨ (81/4 sin⁴(t) cos(t))(3 dt)
Using trigonometric identities, we can simplify the integrand to:
(81/4)(1/5)(sin⁵(t))' = 81/20 sin⁵(t)
Evaluating the integral from t = 0 to t = pi, we get:
∫c xy⁴ ds = ∫₀ᴨ 81/20 sin⁵(t) dt = 81/16π
Therefore, the value of the line integral is 81/16π.
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Which angles are supplementary to 16? Select all that apply.
Answer:
Step-by-step explanation:
15,13,11,9,1,3,5,7
Answer:
13
Step-by-step explanation:
Supplementary angles add up to 180 and angles on a straight line add up to 180 so 13
sheri’s cab fare was $32, with a 20% gratuity and no taxes. sheri's write a check to the cab driver for $40. is this a reasonable amount? explain.
In a case whereby sheri’s cab fare was $32, with a 20% gratuity and no taxes. sheri's write a check to the cab driver for $40, this can be considered as being reasonable amount because it is $1.60 more to the cab driver.
How can we know if it is reasonable?A gratuity is a sum of money that customers typically give to specific service sector employees, including those in the hotel industry, in addition to the service's base charge for the work they have completed.
Given ; Sheri’s cab fare was $32 and the percentage of gratuity is 20%
amount of gratuity = 20% 0f 32 = 6.40
The fare of the cab + gratuity = 32 + 6.40 = 38.40
Check to the cab driver for $40 , implies ($40 - $38.40)= $1.60 more to the cab driver.
Hence, it is reasonable.
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Which of the following is true?
Group of answer choices
(LF – LS) > (EF – ES)
Slack = (LF – LS)
(LS – ES) >= 0
The statement that is true is (LS – ES) >= 0. Option C
How to determine the statementThe latest possible time to begin an activity is denoted as LS in project management, while the earliest possible time is indicated by ES.
The distinction between LS and ES signifies the flexibility an activity possesses, allowing it to be postponed without affecting the ultimate timeframe of the project.
The calculation for Slack involves subtracting the earliest possible finish time (LS) from the latest possible finish time (LF). One must note that (LS - ES) will never be negative as the earliest start time (ES) can never be greater than the latest start time (LS).
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On a 50-point quiz, Jenna earned 82%
of the points. How many points did she
earn?
Answer:
x=41
Step-by-step explanation:
x/50=82/100
cross multiply
100x= 82×50
100x= 4100
x= 4100/100
x= 41
check our work
41/50 × 100= 82 percent
What nominal interest rate compounded monthly is equivalent to
2.50% compounded quarterly? Round to two decimal places
Rounding this to two decimal places, the equivalent nominal interest rate compounded monthly is approximately 2.53%.
To determine the equivalent nominal interest rate compounded monthly for a given nominal interest rate compounded quarterly, we can use the formula for nominal interest rate conversion.
The formula is:
r = (1 + i/n)^n - 1
Where:
r is the nominal interest rate compounded annually (we're looking for this),
i is the nominal interest rate compounded quarterly (2.50% in this case),
n is the number of compounding periods per year (4 for quarterly compounding).
Plugging in the values, we have:
r = (1 + 0.025/4)^4 - 1
Calculating this expression, we find:
r ≈ 0.025335
Rounding this to two decimal places, the equivalent nominal interest rate compounded monthly is approximately 2.53%.
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Can I get help with number 25 its really hard for me.
hi help w/ this question pls i'll give u a brainly
Answer:
Keep adding 8 56 times subtract three which is 472
Step-by-step explanation:
mulitiply 59 times 8
Write a question that matches this equation:
45-n=15
Answer:
ayla had 45 cupcakes for the class jayla gave a number out to the cladd Jayla took 15 cupcakes back home what is the does n equal to
HELPP I WILL GIVE YOU BRAINLIEST!! i think the answer is 42x + 12 but i am not sure!!
Can someone help me with this question. Will Mark brainliest.
Answer:
Step-by-step explanation:
For the given margin of error and confidence level, determine the sample size required. A researcher wishes to estimate the proportion of fish in a certain lake that are inedible due to pollution of the lake. What sample size will ensure a margin of error of at most 0.053 for a 97.5% confidence interval? In the past, similar research determined that 34% of the fish in the lake are inedible.
The sample size required to estimate the proportion of inedible fish in the lake with a margin of error of at most 0.053 and a 97.5% confidence level is 491.
To determine the sample size needed for the estimation, we can use the formula:
n = (Z² * p * q) / E²,where:n = sample size,Z = Z-score corresponding to the desired confidence level (in this case, 97.5% corresponds to a Z-score of approximately 1.96),p = estimated proportion of inedible fish (34% or 0.34),q = 1 - p (probability of fish being edible, which is 1 - 0.34 = 0.66),E = margin of error (0.053).Plugging in these values, we get:
n = (1.96² * 0.34 * 0.66) / 0.053²,n ≈ 491.Therefore, a sample size of 491 will ensure a margin of error of at most 0.053 for a 97.5% confidence interval when estimating the proportion of inedible fish in the lake.
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ou want to put shingles on the outside walls and solar panel the roof of the barn shown. It costs $ for each square meter of shingles. Solar panels cost $ per square meter. How much will this project cost?
4 m
4 m
8 m
9 m
10 m
7 m
5 m
In the accompanying diagram of triangle ABC , D is a point on AC , AB is extended to E, and DE is drawn so that triangle ADE~ triangle ABC . If m
Answer:
m∡ADE = 80°
Step-by-step explanation:
m∡ADE = m∡ABC
m∡B = 180-(30+70) = 80°
therefore, m∡ABC = 80° and so does m∡ADE because they are congruent
True or False: When conducting a survey of a group of people, you must interview every person in that population A Truc B False
Which is a better price: 5 for $1,4
for $0.85, 2 for $0.38, or 6 for
$1.10?
Answer:
6 for $1.1
Step-by-step explanation:
what is the factorization of the polynomial below 16x2-9
Answer:
(4x - 3)(4x + 3)
Step-by-step explanation:
16x² - 9 ← is a difference of squares and factors in general as
a² - b² = (a - b)(a + b) , then
16x² - 9
= (4x)² - 3²
= (4x - 3)(4x + 3)
Use Rouché's Theorem to find the number of (complex) roots, counting multiplicities, of 2z8 + 3z5 - 9z³ + 2 = 0 in the region 1 < |z| < 2.
To apply Rouché's Theorem, we consider two functions: f(z) = 2z⁸ + 3z⁵ - 9z³ + 2 and g(z) = 2z⁸. We want to analyze the number of roots of f(z) = 0 inside the region 1 < |z| < 2.
First, let's examine the behavior of f(z) and g(z) on the boundary of the region. When |z| = 1, the term 2z⁸ dominates over the other terms in f(z), so |f(z)| < |g(z)|. On the other hand, when |z| = 2, the term 2z⁸ is still dominant, and again |f(z)| < |g(z)|.
Since |f(z)| < |g(z)| on the boundary of the region, Rouché's Theorem guarantees that f(z) and g(z) have the same number of roots inside the region, counting multiplicities. In this case, g(z) = 2z⁸ has exactly eight roots, counting multiplicities.
Therefore, by Rouché's Theorem, we can conclude that the equation 2z⁸ + 3z⁵ - 9z³ + 2 = 0 has eight roots, counting multiplicities, inside the region 1 < |z| < 2.
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what value would make the set of points a function. (9, -2) (4,
3) (8,10) (? , 8)
Answer:
Any x-value other than 4, 8, or 9 would make this set of points a function.
1. The three year discount factor is 0.7773 and the one year discount factor is 0.9434. Calculate the two year discount factor if the three year annuity factor is 2.6065 a. 0.9623 b. 0.8758 c. 0.9132
After considering the given data we conclude that the generated two year discount factor is 0.9132, under the condition that three year annuity factor is 2.6065.
Applying the formula for the annuity factor, we can find the two year discount factor:
[tex]annuity factor = (1 - discount factor^n) / r[/tex]
Here,
n = number of years
r = annual interest rate.
We are given that the three year discount factor is 0.7773 and the one year discount factor is 0.9434. We are also given that the three year annuity factor is 2.6065.
Applying the formula for the annuity factor, we can evaluate the annual interest rate:
[tex]2.6065 = (1 - 0.7773^3) / r[/tex]
r = 0.1
Now, we can evaluate the two year discount factor:
[tex]annuity factor = (1 - discount factor^2) / 0.1[/tex]
[tex]2.6065 = (1 - discount factor^2) / 0.1[/tex]
[tex]discount factor^2 = 0.89335[/tex]
discount factor = [tex]\sqrt(0.89335)[/tex]
discount factor = 0.9132
Therefore, the two year discount factor is 0.9132.
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I need help on this question
You are saving money to buy a car. You put $2500 in a savings account that pays 4% annual interest compounded monthly (hint n = 12). A. Write a function that models the amount of the money in the account over time. B. Find the cost of the car after 10 years.
Answer:
FV= $3,726.93
Step-by-step explanation:
Giving the following information:
Initial investment (PV)= $2,500
Interest rate (i)= 0.04/12= 0.003333 monthly
Number of periods (n)= x months
To calculate the future value giving any number of months, we need to use the following formula:
FV= PV*(1 + i)^n
For 10 years:
n=10*12= 120 months
FV= 2,500*(1.003333^120)
FV= $3,726.93
Gina had 1/2 a liter of Dr. Pepper. She gave 2/5 of a liter to her friend. How much does she have left?
Answer:
1/10
Step-by-step explanation:
1) find a common factor between the two denominators.
2 and 5 have a common factor of 10. you can find this by multiplying the denominators.
2) now that you have a common denominator of 10, you'll need to change the numerators too. you do this by finding out how many times the denominator goes into your common factor. 5 goes into 10 two times so you would multiply 2 x 2. two goes into 10 five times so multiply 5 x 1.
3) you now have both fractions that share a common factor. now you have to subtract them.
5/10 - 4/10 = 1/10