1. Suppose A
2.Then B is bounded below.
3. Let x = lub(A).
4. Then -x = glb(B).
a) Explain why (2) is true.
b) Explain why lub(A) exists.
c) Explain why (4) is true.
d)Deduce that if B
Answers
Hence, −u = glb(A) = lub(−A), and u + x = sup(C − A) = lub(C) − glb(A).
a) Bounded below means that there is a number x such that for all y in B, x ≤ y. Since x is an upper bound of A, it follows that x is a lower bound of B. Hence, B is bounded below.
b) Any non-empty set of real numbers that is bounded above has a least upper bound. Since A is bounded above (by any upper bound of B, for instance), it follows that A has a least upper bound.
c) By the definition of the least upper bound, x is an upper bound of A, and for any ε > 0, there exists a ∈ A such that x − ε < a. Since x is an upper bound of A, it follows that −x is a lower bound of B. By a), B is also bounded below, hence it has a greatest lower bound. Let y = glb(B). Then for any ε > 0, there exists b ∈ B such that b < y + ε, which implies that −(y + ε) < −b. Since −x is a lower bound of B, it follows that −x ≤ −b for all b ∈ B, hence −x ≤ y + ε for all ε > 0. Thus, −x ≤ y.
d) Suppose B is non-empty and bounded above, and let z = sup(B). Then for any ε > 0, there exists b ∈ B such that b > z − ε. Since x is the least upper bound of A, there exists a ∈ A such that a > x − ε. Then a + b > x − ε + z − ε = (x + z) − 2ε. Since ε was arbitrary, it follows that x + z is the least upper bound of the set C = {a + b | a ∈ A, b ∈ B}. In particular, C is non-empty and bounded above, hence it has a least upper bound. Let w = lub(C), and let ε > 0 be given. Then there exist a ∈ A and b ∈ B such that a + b > w − ε. Since x is the least upper bound of A, there exists a' ∈ A such that a' > x − ε. Then a' + b > w − ε + ε = w, which implies that w is an upper bound of C. By the definition of the least upper bound, it follows that w ≤ x + z. Since −x = glb(B), it follows that −x ≤ z, hence w ≤ x − (−x) = 2x. But x + z ≤ 2x, hence w ≤ x + z ≤ 2x. Since x is an upper bound of A, it follows that −x is a lower bound of −A, hence by a), −A is bounded below. Let u = glb(−A). Then u + x = glb(C − A), where C − A = {b − a | a ∈ A, b ∈ B}. But B is bounded below, hence C − A is also bounded below, and glb(C − A) exists. Hence, u + x is the greatest lower bound of C − A. Let ε > 0 be given. Then there exist a ∈ A and b ∈ B such that a + b < u + x + ε. Since u is the greatest lower bound of −A, it follows that −a > −u. Then b − (u + a) < x + ε, hence b − (u + a) < ε. Since ε was arbitrary, it follows that u + x is an upper bound of C − A. By the definition of the least upper bound, it follows that u + x = sup(C − A). Hence, −u = glb(A) = lub(−A), and u + x = sup(C − A) = lub(C) − glb(A).
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Related Questions
Graph the inequality
4x-12y<24
Show all your work!
Answers
Answer:
Slope = 0.667/2.000 = 0.333
x-intercept = 6/1 = 6.00000
y-intercept = 6/-3 = 2/-1 = -2.00000
Step-by-step explanation:
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
4*x-12*y-(24)=0
Step 1: Pulling out like terms
Pull out like factors :
4x - 12y - 24 = 4 • (x - 3y - 6)
Equation at the end of step 1:
Step 2:
Equations which are never true:
Solve : 4 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Equation of a Straight Line
Solve x-3y-6 = 0
Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).
"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.
In this formula :
y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis
The X and Y intercepts and the Slope are called the line properties. We shall now graph the line x-3y-6 = 0 and calculate its properties
Graph of a Straight Line :
Calculate the Y-Intercept :
Notice that when x = 0 the value of y is 2/-1 so this line "cuts" the y axis at y=-2.00000
y-intercept = 6/-3 = 2/-1 = -2.00000
Calculate the X-Intercept :
When y = 0 the value of x is 6/1 Our line therefore "cuts" the x axis at x= 6.00000
x-intercept = 6/1 = 6.00000
Calculate the Slope :
Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is -2.000 and for x=2.000, the value of y is -1.333. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of -1.333 - (-2.000) = 0.667 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)
Slope = 0.667/2.000 = 0.333
Geometric figure: Straight Line
Slope = 0.667/2.000 = 0.333
x-intercept = 6/1 = 6.00000
y-intercept = 6/-3 = 2/-1 = -2.00000
6х + Зу = 12
Slope-intercept
Answers
Answer:
x=2,y=4
Step-by-step explanation:
Solve the following absolute value equation. |2+1 = 3 3 2 = -[ ]
Answers
Answer:
Sorry
Step-by-step explanation:
I not know this one
Provide a detailed distinction between the two training
methodologies that may be utilised in the organisation, as well as
the popular methods for each.
Answers
In an organization, two popular training methodologies that can be utilized are on-the-job training and off-the-job training.
Let's examine each methodology along with their popular methods:
1. On-the-Job Training:
On-the-job training refers to the process of acquiring knowledge and skills while performing tasks and responsibilities directly related to one's job. It takes place within the actual work environment and provides hands-on experience. Some popular methods of on-the-job training include:
a. Job Shadowing: This method involves a new employee observing and working closely with an experienced employee, learning from their day-to-day activities and tasks.
b. Mentoring: In this method, a more experienced employee (mentor) guides and supports a less experienced employee (mentee) by providing advice, feedback, and sharing their expertise.
c. Coaching: Similar to mentoring, coaching involves a more experienced employee providing guidance and support to enhance the skills and performance of the employee being coached.
d. Apprenticeships: This method combines on-the-job training with classroom instruction. It allows individuals to learn a trade or skill through practical experience while working alongside skilled professionals.
2. Off-the-Job Training:
Off-the-job training refers to the process of learning and acquiring skills outside of the regular work environment. It usually takes place in a separate training facility or classroom setting. Some popular methods of off-the-job training include:
a. Workshops and Seminars: These are interactive sessions where participants learn new skills, gain knowledge, and exchange ideas on specific topics facilitated by subject-matter experts.
b. Conferences and Conventions: These events bring together professionals from a particular industry or field to attend presentations, panel discussions, and networking opportunities to learn about new trends and developments.
c. E-Learning and Online Courses: With advancements in technology, online platforms offer various courses and training programs that individuals can access remotely at their own pace and convenience.
d. Simulations and Role-Playing: These methods involve creating artificial scenarios or situations that closely resemble real work situations, allowing participants to practice and develop skills in a controlled environment.
e. Case Studies and Group Discussions: Participants analyze real-life scenarios and engage in group discussions to understand problem-solving techniques, decision-making processes, and best practices.
It's important to note that the selection of the training methodology and methods depends on the specific needs, goals, and resources of the organization, as well as the desired learning outcomes for the employees. A combination of both on-the-job and off-the-job training can often be beneficial for comprehensive skill development and knowledge acquisition.
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EFGH is a parallelogram. Find the measure of JG.
Answers
\( \large{ { \tt{꧁ \: A\: N \: S \: W \: E \: R \: ꧂}}}: \)
Step-by-step explanation:
✧ \( \underline{ \underline{ \large{ \red{\tt{ \: P \: R \: O \: V \: I\: D \: E \: D\: }} }}: }\)
EFGH is a parallelogram.EF = 3w & JG = w + 8✎ \( \underline{ \underline{ \purple{ \large\tt{T\: O \: \: F \: I\: N \: D}}}} : \)
measure of JGSince we know , The diagonals of a parallelogram bisect each other ;
\( \large{\text{EF \: = JG}}\)
➝ \( \large{ \bf{3w = w + 8}}\)
➝ \( \large{ \bf{3w - w = 8}}\)
➝ \( \large{ \bf{2w = 8}}\)
➝ \( \large{ \bf{w = \frac{8}{2}}} \)
➝ \( \boxed{\large{ \bf{w = 4}}}\)
♔ \( \underline{ \text{Replacing \: Value}} : \)
⇢\( \large{ \bf{JG = w + 8 = 4 + 8 = \boxed{ \bf{12}}}}\)
Yayy! We found our answer ☾
⛧ \( \boxed{ \boxed{ \orange{\tt{Our \: Final \: Answer : \underline{ \boxed{ \tt{12}}}}}}}\)
Hope I helped! ♡
Have a wonderful day / night ! ☘
Let me know if you have any queries regarding my answer ♬
⋆ \( \underbrace{ \overbrace{ \mathfrak{Carry \:On \: Learning}}}\) ⋆
▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁
Given: a and b are parallel and c is a transversal.
Prove: ∠2 ≅ ∠7
Parallel lines b and a are cut by transversal c. On line b where it intersects with line c, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 1, 5, 6, 2. On line a where it intersects with line c, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 3, 7, 8, 4.
Use the drop-down menus to complete the paragraph proof showing that alternate interior angles are congruent.
We know that lines a and b are parallel and that line c is a transversal because that is given. We can tell that angles 2 and 5 are congruent because
angles are congruent. Angles 5 and 7 are congruent because angles by parallel lines cut by a transversal are congruent. Therefore, angles 2 and 7 are congruent based on the
.
Answers
The missing words to complete the proof are respectively; Vertical Angles; Corresponding angles; Transitive Property
How to prove congruent angles?The image of the transversal line is attached.
1) We know that lines a and b are parallel and that line c is a transversal because that is given.
2) We can tell that angles 2 and 5 are congruent because vertical angles are congruent.
3) Angles 5 and 7 are congruent because corresponding angles by parallel lines cut by a transversal are congruent.
4) Therefore, angles 2 and 7 are congruent based on the transitive property.
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Answer: vertical, corresponding, and transitive property
Step-by-step explanation: I got the correct answer on Edge 2022. The answer is correct trust me! Proof is shown down below.
help meeeeeeeeee pleaseeeeeeeeeeee!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
please help me !!!!!!!!
Answers
the correct answer is J
hope this helps!
In how many ways can the digits in the number 9550000 be arranged
Answers
The number 9550000 can be arranged in a total of 8!/(4!4!) = 70 different ways.
What is digits?Digits are numerals or numbers used for counting, measuring, and representing quantities. They are the building blocks of mathematics, and are used to express amounts, relationships, and orders of magnitude. Digits are also used in computers and other digital devices to store and communicate information.
The 8 digits of 9550000 can be arranged using the formula for permutation: nPr = n!/(n-r)!
In this case, n = 8 and r = 8. So, the number of permutations of the 8 digits of 9550000 will be 8!/(8-8)! = 8!/(0!) = 8! = 40320.
However, we need to account for the fact that there are 4 zeros in the number 9550000. To do this, we need to divide the number of permutations by 4! (factorial of 4) to account for the 4 identical digits.
Therefore, the total number of ways the digits in the number 9550000 can be arranged is 40320/4! = 70.
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The measure of angle is 14 degrees. Find the measure of angle P.
Answers
Answer:
32
Step-by-step explanation:
74+74=148
180-148=32
A standardized test is designed so that scores have a mean of 50 and a standard deviation of 4. What percent of scores are between 46 and 54?
Answers
Answer:
c
Step-by-step explanation:
c is correct
$1.66 x 100 = ? I can't seem to find the correct answer and I must show my work
Answers
Answer:
166
Step-by-step explanation:
Answer:
$1.66x100 would be 166 or $166.
Step-by-step explanation:
Since you are multiplying by 100, you could move the decimal point to the right twice which would give you $166, or you could multiply normally.
Please help this is my last one
Answers
Answer:
2 is corresponding angles
3 is alternate interior angles
4 is sum of angles
PLS HELP ITS MATH AND I DONT UNDERSTAND
No links pls and ty <3
Answers
C=52.08π
Solution:Circumference of a circle:C=2πrwhereπ=3.14...r= radiusIn this case, r is equivalent to 26.04 ft.C=2π×26.04Multiply:C=52.08π (in terms of pi)Hope it helps.
Do comment if you have any query.
Answer:
The last option \(52.08\pi\) feet
Step-by-step explanation:
\(c=2\pi r=2\pi (26.04)=52.08\pi\)
Hope this helps
Please answer question C . What's the rule in the number chain??
Answers
The rule that could be used to find the next number in item b is given as follows:
x 3.
What is a geometric sequence?A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.
In item b, we have that each term is the previous term multiplied by 3, hence the common ratio is given as follows:
q = 3.
Thus the rule that could be used to find the next number in item b is given as follows:
x 3.
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7(2x+3)=3(4x+6)+2x+3
Answers
Answer:
\(7(2x + 3) = 3(4x + 6) + 2x + 3 \\ 14x + 21 = 12x + 18 + 2x + 3 \\ 14x - 14x = 21 - 21 \\ 0 = 0\)
Which represents the reference angle for 2pi/3
Answers
Answer:
The reference angle of \(\frac{2\pi}{3}\) is \(\frac{\pi}{3}\)
Step-by-step explanation:
Given
\(\frac{2\pi}{3}\)
Required
Which represent its reference angle
The very first step is to determine the quadrant of the given angle
For better understanding;
Convert \(\frac{2\pi}{3}\) to degrees
\(\frac{2\pi}{3}\) = \(\frac{2}{3} * 180\)
\(\frac{2\pi}{3}\) = \(\frac{2 * 180}{3}\)
\(\frac{2\pi}{3}\) = \(\frac{360}{3}\)
\(\frac{2\pi}{3}\) = \(120\)
120 degrees in in the second quadrant; Hence, \(\frac{2\pi}{3}\) is in the second quadrant
The reference angle of \(\frac{2\pi}{3}\) is calculated using the following formula;
\(\pi - \frac{2\pi}{3}\)
Rewrite \(\pi\)
\(\pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3}\)
Take LCM
\(\pi - \frac{2\pi}{3} = \frac{3\pi - 2\pi}{3}\)
\(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\)
Hence, the reference angle of \(\frac{2\pi}{3}\) is \(\frac{\pi}{3}\)
The reference angle for the angle 2π/3 is, π/3
Given that,
An angle is,
⇒ 2π/3
To determine the reference angle we need to find the quadrant of the given angle.
For this, we can convert angle into degree,
2π / 3 = 2 × 180 / 3
= 120°
Since, 120 degrees is lies in the second quadrant.
Hence, 2π/3 is in the second quadrant.
So, The reference angle is,
⇒ π - 2π/3
⇒ π/3
Therefore, The reference angle for the angle 2π/3 is, π/3
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A ball is dropped from a height of 384 feet. If it rebounds 3 4 of the height from which it falls every time it hits the ground, how high will it bounce after it strikes the ground for the fourth time
Answers
After striking the ground for the fourth time, the ball will bounce to a height of approximately 243 feet.
When the ball is dropped from a height of 384 feet, it rebounds to 3/4 of the height from which it falls. This means that after the first bounce, the ball reaches a height of (3/4) 384 = 288 feet. On the second bounce, it reaches (3/4) 288 = 216 feet. On the third bounce, it reaches (3/4) 216 = 162 feet. Finally, on the fourth bounce, it reaches (3/4) 162 = 121.5 feet.
However, it's important to note that the question asks for the height after the ball strikes the ground for the fourth time, not the height after the fourth bounce. Each bounce consists of a fall and a rise, so the ball strikes the ground twice in each bounce. Therefore we will use geometric sequence formula, the ball has struck the ground three times already (after the first three bounces), and it will strike the ground once more before reaching its final height. Thus, the ball will bounce to a height of approximately 121.5 + 121.5 = 243 feet after striking the ground for the fourth time.
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Em uma sala de aula ,foi realizada uma pesquisa, a qual apontou que 30 alunos gostam de participar esportes.qual e a porcentagem de alunos que gostam de esportes
Answers
60% of the students in the classroom like to participate in sports based on the survey results.
In a classroom survey, it was found that out of a total of 50 students, 30 students expressed their liking for participating in sports. We can now calculate the percentage of students who like sports using the following formula:
Percentage = (Number of students who like sports / Total number of students) × 100
Plugging in the values from our survey:
Percentage = (30 / 50) × 100 = 60%
Therefore, 60% of the students in the classroom like to participate in sports based on the survey results.
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The complete question is-
In a classroom, a survey was conducted that showed that 30 students out of a total of 5 students like to participate in sports. What percentage of students like sports?
Which is the better buy? A pack of 6 pens for $1.30 or a pack of 40 pens for $7.50? Find the unit rate
Answers
Answer:
40 pens for $7.50 is a better deal because it is about 19 cents per pen while 6 for $1.30 is 22 cents per pen
Step-by-step explanation:
1.30/6=0.22 rounded
7.50/40=0.19 rounded
The vera family is traveling from Philadelphia to Orlando for vacation the equation Y equals 1000-65X represents the distance in miles remaining in the trip after X hours interpret the slope and the Y intercept
Answers
As per the following graph, the slope is 66.67 and the y intercept is 1000.
Graph:
The graph is used to present data that are too numerous or complicated to be described adequately in the text and in less space.
Given,
The Vera family is traveling from Philadelphia to Orlando for vacation the equation Y = 1000-65X represents the distance in miles.
Here we need to find the slope and the y intercept.
For the given equation, we have to plot the graph to identify the y intercept.
If we plot the given equation then we get the following graph.
Through the graph we have identified that, the value of y intercept is 1000.
Then the next thing we have to do is to find the slope of the lien for that we have to take the two sample point from the graph.
They are,
(0,1000) and (6,600)
Then the slope of the line is calculated through the formula,
m = (600 - 1000)/6 - 0
m = -400/6
m = 66.67
Therefore, the slope of the line is 66.67.
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Which choice lists the numbers in increasing order?
Answers
Answer:
D
Step-by-step explanation:
Because negative is smaller than fractions. and ABC, is all off.
Answer: I think the answer is D/ the last answer
Step-by-step explanation:
suppose 1000 people enter a chess tournament. the champion is determined in such a way that a player is eliminated after one loss and games are played until only one entrant has not lost.how many games must be played to determine a champion?
Answers
999 games must be played to determine a champion in a tournament with 1000 entrants.
The number of games that must be played to determine a champion in a tournament with 1000 entrants, we can use the concept of a binary tree.
In the first round, there will be 1000 players, and 500 games will be played (as each game eliminates one player).
In the second round, there will be 500 players remaining, and 250 games will be played.
In the third round, there will be 250 players remaining, and 125 games will be played.
This process continues until there is only one player remaining, who becomes the champion.
The total number of games played, we can sum the number of games played at each round
500 + 250 + 125 + 62 + 31 + 15 + 7 + 3 + 1 + 5 = 999
Therefore, 999 games must be played to determine a champion in a tournament with 1000 entrants.
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we know that for a probability distribution function to be discrete, it must have two characteristics. one is that the sum of the probabilities is one. what is the other characteristic?
Answers
The other characteristic of a discrete probability distribution function is that each individual outcome has a probability greater than or equal to zero.
In other words, the probability assigned to each possible value in the distribution must be non-negative. This ensures that the probabilities are valid and that the distribution accurately represents the likelihood of each outcome occurring. So, the two characteristics of a discrete probability distribution function are: (1) the sum of the probabilities is one, and (2) each individual outcome has a probability greater than or equal to zero.
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Tell me at least 3 distinct things you know about i and imaginary numbers.
Answers
i is equal to _/-1
I NEED HELP WITH THESE RNNNN!!!!!!!
Answers
Answer:
Step-by-step explanation:
i really need pleaseee asap !!!
Answers
The solution is (6,-2)
do the following study results require a post-hoc test to be performed? when testing four groups, it was found that exercise does not affect memory f(3,26)1.92,p>.05 yes no
Answers
Yes, the study results require a post-hoc test to be performed.
Since the main analysis, an ANOVA test, showed a non-significant result (F(3,26) = 1.92, p > .05), it may be tempting to conclude that there is no difference among the four groups. However, to ensure the accuracy of the findings, a post-hoc test should be conducted.
A post-hoc test is necessary because it helps to identify if there are any specific pair-wise differences among the groups that were not detected by the initial ANOVA test. Although the overall result may not be significant, there might still be significant differences between specific group pairs.
By conducting a post-hoc test, you can reduce the risk of Type II errors (false negatives) and better understand the underlying relationships between exercise and memory in the study. Some popular post-hoc tests include Tukey's HSD, Bonferroni, and Scheffe tests.
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"if i purchase this product for $79.99 and two accessories for $9.99 and $7.00, how much would i owe after the 8.75% tax is applied?" employee: "your total would be __________."
Answers
Answer:$96.98
Step-by-step explanation: all you just do is to add it all up to the total amount.
Answer: $105.47
I hope you get the job!!!!!
There are 300 students in the 9th grade and each student is assigned a locker. The 300 students start down the hall one at a time. The first student opens every locker. The second student closes all the lockers that are multiples of 2. The third student changes the lockers that are multiples of 3 (which means if a locker is closed, they open it. If the locker is open, they close it.) The fourth student changes all the lockers that are multiples of 4. This continues until the 300th student changes the 300th locker. Then, the principal walks down the hall, and changes all the lockers that are prime numbers. At the end, how many lockers will be closed?
Answers
Answer:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289
Step-by-step explanation:
The pattern will lead to the answer being all the perfect squares from 1 to 300.