Step-by-step explanation:
distinct ways !
STATICS and STATICS are 2 different ways to arrange them. have you noticed the difference ?
no, of course not. I only exchanged the 2 S. the same problem with the 2 T.
while they are different permutations, their arrangements are not distinctly different. to us observers they look the same.
so, for the 7 letters, 7! is not the full answer.
these 7! permutations include all the ones, where we only change the S and the T with each other.
we need to eliminate those. we do this by dividing by the S permutations (2! = 2) and the T permutations (again, 2! = 2).
so, the answer is
7! / (2×2) = 5040 / 4 = 1260
the 7 letters can be arranged in 1260 distinctly different ways.
find the ear in each of the following cases: input area: output area: stated rate (apr) compounding periods per year effective rate (ear) 7.80% 4 15.30% 12 12.40% 365 11.40% infinite
The effective annual rate (EAR) is the actual rate of interest that is earned or paid over a given period of time, taking into account the compounding of interest. For example, a stated rate of 7.80% with 4 compounding periods per year would result in an EAR of 15.30%.
The effective annual rate (EAR) is a measure of the actual rate of interest that is earned or paid over a given period of time. It takes into account the effect of compounding, which is the process of earning interest on both the principal and the accumulated interest. This means that the rate of return can increase over time. The EAR is calculated by taking the stated rate of interest, which is the rate of interest stated by the lender, and multiplying it by the number of compounding periods per year. For example, if the stated rate is 7.80% and the compounding periods per year are 4, then the EAR would be 15.30%. Similarly, if the stated rate is 12.40% and the compounding periods per year are 365, then the EAR would be 11.40%. If the compounding period is infinite, then the EAR will be the same as the stated rate, in this case 12.40%. Therefore, the EAR is the actual rate of return that is earned or paid over a given period of time.
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the statistical methods of analysis of variance assume that the populations are normally distributed.A. TrueB. False
The statement of statistics "the statistical methods of analysis of variance assume that the populations are normally distributed" is true. So the correct answer is (a)True.
What is statistics?Statistics is a branch of applied mathematics that involves collecting, describing, analyzing, and drawing conclusions from quantitative data. It gives us information to figure out how things work. Statistics are used to conduct research, evaluate results, develop critical thinking, and make informed decisions. Statistics can be used to examine nearly any area of research to find out when and why things happen, and whether their recurrence is predictable.
Who uses statistics?Statistics are used in many different applications and professions. Statistics are generated each time data is collected and analyzed. This ranges from government agencies to academic research to investment analysis.
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Jo has a 15 GB monthly cell phone plan. Jo has used 4 GB in the first 20 days of the month. At that rate, how many GB will Jo use after a month of 30 days?
Answer:
6 GB
Step-by-step explanation:
First find the rate per day. You can do that by dividing 4GB by 20 days.
4/20 = 1/5 = 0.2GB per day
Then, just multiply 30 days by the rate of 0.2GB/day.
30 x 0.2 = 6
the pair of linear equations is graphed below
approximatly 31,500 citizens of a country died in automobile accidents in 2015, express this toll in deaths per day. round to the nearest whole number. There are approximatly how many deaths due to automobile accidents
Using simple division we know that 86 people or citizens dies each day in 2015.
What is division?One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division.
The other operations are multiplication, addition, and subtraction.
Division in mathematics is the process of dividing an amount into equal parts.
For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on.
So, we know that:
31,500 people died in 2015.
We also know that 2015 had 365 days.
Then, people died easily day would be:
= 31,500/365
= 86.30
Rounding off: 86 people
Therefore, using simple division we know that 86 people or citizens dies each day in 2015.
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Determine the tip percentage on a restaurant bill, given a subtotal of $41.00 and a total of $49.20. (2 points)
18%
25%
15%
20%
Using the table, what is the average daily balance of the credit card for the January 1 - January 31 billing period? Round your answer to the nearest cent. Do not include a dollar sign or comma in your answer. For example, $5,678.00 should be entered as 5678.00. Day 1112131 Activity − Purchase Purchase Purchase Adjustment −+200+250+400 Closing Balance 800100012501650
The average daily balance of the credit card for the January 1 - January 31 billing period is $1283.87.
What is the average?In maths, the average value in a set of numbers is the middle value, calculated by dividing the total of all the values by the number of values. When we need to find the average of a set of data, we add up all the values and then divide this total by the number of values.
The average daily balance can be determined by dividing the total daily balances for January by the number of days in January.
The average daily balance = Total balance / Total number of days in January.
Total balance = (800 x 1 days) + (1000 x10 days) + (1250 x 10 days) + (1650 x 10 days)
Total balance = 800 + 10,000 + 12,500 + 16,500 = 39800
The average daily balance = 39800/31 = $1283.87
Therefore, the average daily balance of the credit card for the January 1 - January 31 billing period is $1283.87.
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Here is my question please show all of your work and answer as well
A shipment of 12 X-Ray machines includes 4 that are defective. In how many ways can a hospital purchase 5 of these machines and receive at least 2 of the defective machines
Answer:
Step-by-step explanation:
The number of ways the hospital can purchase 5 machines and receive at least 2 of the defective machines is calculated as follows:
The number of ways the hospital can receive exactly 2 defective machines is C(4,2) = 6, where C(n, k) is the number of combinations of n things taken k at a time.
The number of ways the hospital can receive exactly 3 defective machines is C(4,3) = 4.
The number of ways the hospital can receive exactly 4 defective machines is C(4,4) = 1.
So, the total number of ways the hospital can purchase 5 machines and receive at least 2 of the defective machines is 6 + 4 + 1 = 11.
A box contains 16 square chips, 12 triangular chips, and 4 circular chips. Identify the ratio of circular chips to square chips in all three forms.
The ratio of circular chips to square chips is 1:4.
What is a ratio?The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.
Given:
A box contains 16 square chips, 12 triangular chips, and 4 circular chips.
The ratio of circular chips to square chips,
= Circular chips / square chips
= 4 / 16
= 1/4
= 1: 4
Therefore, the ratio is 1:4.
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I already graphed it but I need yo figure out the solution. PLEASE HELP:)
I will give you brainlest!!!!
The solution of the system of equations is equal to (3, 4).
What is a point of intersection?In Mathematics, a point of intersection can be defined as the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.
By critically observing the graph of the lines representing the given system of equations, we can reasonably and logically deduce that the correct solution set lies in Quadrant II and it is denoted by the point of intersection of both the x-coordinate (x-axis) and y-coordinate (y-axis), which is given by ordered pair (3, 4).
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Describe as precisely thoroughly as possible why critical values of a function f(x) occur either where f′(x) = 0 or where f′(x) is undefined.
Well, most books define critical values of a function as x-values where f'(x)=0 or f'(x) is undefined, so they occur there because that's the definition of what a critical value is.
But the reason we define it that way is that when you're looking for local extrema (local mins and/or local maxs), these must occur when f'(x)=0 or f'(x) is undefined.
If f has a local extremum at a point x=c, then f'(c)=0 or f'(c) is undefined.
for a certain kind of plaster work, 1.1 cu yd of sand are needed for every 100 sq yd of surface. How much sand will be needed for every 100 sq yd of surface. how much is sand will be needed for 380 sq yd of surface
The amount of sand needed for every 100 square yard of surface would be 1. 1 cubic yards.
The amount of sand needed for 280 square yards would be 4 .18 cubic yards of sand.
How to find the sand needed ?The question says that 1. 1 cubic yards of sand are needed for every 100 square yards of surface so this is the amount of sand needed per 100 square yards.
As for 380 square yards, the amount of sand needed can be found by the formula :
= ( 380 square yards of surface x 1. 1 cubic yards of sand ) / 100 square yards of surface
= 418 / 100
= 4 .18 cubic yards of sand
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Determine the range, mean, variance, and standard deviation of the sample data set.
12,17,23,15,15,18,19,17,14,15
Statistics is a discipline, principally within applied mathematics, concerned with the systematic study of the collection, presentation, analysis, and interpretation of data.
The measures used in statistics are mean, range, standard deviation, variance etc.
The range can be calculated by finding the difference between the lowest and highest observation i.e. 23 - 12 = 11.
Variance is a measure of dispersion of data points from the mean. In the above given data, the variance is 9.389.
Standard deviation is the square root of the variance, hence, is 3.06.
Mean = sum of data divided by no of observations (10).
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Zak has 18 tennis balls. This i
Matt has 9 more tennis balls
How many tennis balls does
Answer:
The cost of 3 tennis balls and 2 golf balls is $7. The cost of 6 tennis balls and 3 golf balls is$12. Find the cost of 1 tennis ball and 1 golf ball.
Step-by-step explanation:
Question 9 of 12
In the straightedge and compass construction of the congruent angle below,
which of the following reasons can you use to prove that LABC = LFDE?
D
B
E
C
A. Draw a line through points A and Cand another line through points
Fand E. Then AABC AFDE by SSS, and ZABC LFDE by
CPCTC.
B. Draw a line through points A and Cand another line through points
F and E. Then AABC AFDE by SSA, and LABC = LFDE by
CPCTC.
C. Draw a line through points A and C and another line through points
Fand E. Then AABC AFDE by AAS, and LABC LFDE by
CPCTC.
D. Draw a line through points A and Cand another line through points
F and E. Then AABC AFDE by ASA, and LABC LFDE by
CPCTC.
The correct option that can be used to prove that ∠ABC = ∠FDE is option (A). It is done using rules of congruency.
What is meant by congruency?
In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.
For the construction of the congruent angle, the compass measurement used is the same for both figures.
Now we draw a line through points A and C and another line through points F and E.
We can then say that
AB = FD
BC = DE
CA = EF
These are the sides of the triangles ΔABC and ΔFDE
As all sides are equal, we can say that
ΔABC ≅ ΔFDE by SSS congruency rule.
Now CPCTC rule says that "corresponding parts of congruent triangles are congruent". This means that the corresponding angles and sides of two or more triangles must also be identical in order for them to be considered congruent.
So ∠ABC ≅ ∠FDE by CPCTC as they are corresponding angles.
Therefore the correct congruency rules that can be used to prove that
∠ABC = ∠FDE is option (A).
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Find the derivative of the following function. S = t^23 ln|t| Which rule of differentiation will be most helpful in beginning this problem? A. The chain rule B. The product rule C. The sum or difference rule D. The constant rule E. The power rule The expression in |t| appears as a subexpression of the given function. What is the derivative of this expression? d/dt (ln|t|) = t/|t|^2 Find the derivative of the given function ds/dt =
The derivative of the given function is 23t^22 ln |t| + t^22
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a complete sentence. Any one of the following mathematical operations can be used. A sentence has the following structure: Number/variable, Math Operator, Number/Variable is an expression.
The product rule will be most helpful in beginning this problem.
S = t^23 ln|t|
Differentiate with respect to t
D(s) = ds/dt = D( t^23 ln|t| )
Apply the rule :
D(uv) = u'v+uv'
u' =du/dt , v'=dv/dt
D(s) = 23t^22 ln |t| + t^23 (1/t)
D(s) = 23t^22 ln |t| + t^22
The derivative of the given function is 23t^22 ln |t| + t^22
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The derivative of the given function is 23t²² ln |t| + t²².
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a complete sentence. Any one of the following mathematical operations can be used. A sentence has the following structure: Number/variable, Math Operator, Number/Variable is an expression.
The product rule will be most helpful in beginning this problem.
S = t²³ ln |t|
Differentiate with respect to t
D(s) = ds/dt = D( t²³ln|t| )
Apply the rule :
D(uv) = u'v+uv'
u' =du/dt , v'=dv/dt
D(s) = 23t²² ln |t| + t²³ (1/t)
D(s) = 23t²² ln |t| + t²²
The derivative of the given function is 23t²² ln |t| + t²²
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Please solve this for me, no other information was given with this question
The percentages of the weights are given as follows:
a) Over 15.5 pounds: 2.07%.
b) Under 11.9 pounds: 75.8%.
c) Between 11.9 pounds and 15.9 pounds: 22.13%.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The parameters for this problem are given as follows:
[tex]\mu = 10, \sigma = 2.7[/tex]
The proportion over 15.5 pounds is one subtracted by the p-value of Z when X = 15.5, hence:
Z = (15.5 - 10)/2.7
Z = 2.04
Z = 2.04 has a p-value of 0.9793.
1 - 0.9793 = 0.0207.
The proportion under 11.9 pounds is the p-value of Z when X = 11.9, hence:
Z = (11.9 - 10)/2.7
Z = 0.7
Z = 0.7 has a p-value of 0.7580.
The proportion between is the subtraction of the p-values, hence:
0.9793 - 0.7580 = 0.2213 = 22.13%.
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y = x - 11
7x+7y=-21
systems of equations by substitution
Answer: We can solve the system of equations using the substitution method.
First, we'll isolate "y" in one of the equations and substitute it into the other equation:
Y = x - 11
7x + 7y = -21
Substituting the first equation into the second equation:
7x + 7(x - 11) = -21
7x + 7x - 77 = -21
14x = 56
x = 4
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y:
Y = x - 11
Y = 4 - 11
Y = -7
The solution to the system of equations is (x, y) = (4, -7).
Step-by-step explanation:
The value of x and y in the equation is 4 and -7 respectively
What is Simultaneous equation?Simultaneous equations are two or more algebraic equations that share variables e.g. x and y . They are called simultaneous equations because the equations are solved at the same time. For example, below are some simultaneous equations: 2x + 4y = 14 4x − 4y = 4.
Using substitution method to solve the system,
y = x-11 equation 1
7x+7y = -21 equation 2
substitute x-11 for y in equation 2
7x + 7( x-11) = -21
7x + 7x -77 = -21
14x = -21+77
14x = 56
divide both sides by 14
x = 56/14
x = 4
substitute 4 for x in equation 1
y = 4-11
y = -7
therefore the value of x and y is 4 and -7 respectively
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Find the length of the third side. If necessary, round to the nearest tenth. 9 12
The required length of the third side of the right-angle triangle is 15 units.
What are Pythagorean triplets?In a right-angled triangle, its sides, such as the hypotenuse, are perpendicular and the base is Pythagorean triplets.
Here,
For the right triangle lenght of the two legs to be 9 and 12 units, a measure of the third side is to be determined.
So, Applying the Pythagoras theorem,
h² = p² + b²
h = √9² + 12²
h = √225
h = 15
Thus, the required length of the third side of the right-angle triangle is 15 units.
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Which is the best estimate of the difference between 5 5/6 and 3 1/4
The best estimate of the difference between the two numbers is 31/12 or 2 7/12.
What are mixed fractions?A mixed fraction is one that is represented by both its quotient and remainder. A mixed fraction is, for instance, 2 1/3, where 2 is the quotient and 1 is the remainder. An amalgam of a whole number and a legal fraction is a mixed fraction.
to transform a mixed fraction into an incorrect fraction.
By denominator, divide the numerator.
Take the quotient as a whole number and use the remaining amount as the proper fraction's numerator while maintaining the original denominator.
Covert the mixed fraction into improper fractions:
5 5/6 = 35/6
3 1/4 = 13/4
The difference between the two numbers is:
35/6 - 13/4
Take the LCM.
70/13 - 39/12
(70 - 39) / 12
31/12 = 2 7/12
Hence, the best estimate of the difference between the two numbers is 31/12 or 2 7/12.
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Find fo g, go f, and g o g.
f(x) = 2x, g(x) = x²
(a)
fog
2x4
(b) gof
16x4
(c) gog
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The required values of the given composite functions follow as:
(a) fog(x) = 4x², (b) gof(x) = 2x², and (c) gog(x) = x⁴.
What are composite functions?Composite functions are mathematical functions that are formed by taking the output of one function and using it as the input for another function.
(a) To find fog(x), we need to evaluate g(f(x)).
Since function f(x) = 2x, we have:
g(f(x)) = g(2x) = (2x)² = 4x²
Therefore, fog(x) = 4x².
(b) To find gof(x), we need to evaluate f(g(x)).
Since g(x) = x², we have:
f(g(x)) = f(x²) = 2(x²) = 2x²
Therefore, gof(x) = 2x².
(c) To find gog(x), we need to evaluate g(g(x)).
Since function g(x) = x², we have:
g(g(x)) = g(x²) = (x²)² = x⁴
Therefore, gog(x) = x⁴.
Thus, the required values of the given composite functions follow as:
(a) fog(x) = 4x², (b) gof(x) = 2x², and (c) gog(x) = x⁴.
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PLEASE HELP DUE IN FEW MINUTES ASAP THANK YOU
Answer:
[tex]\frac{(s + 15.4)(8.1)}{2}[/tex] = 81
8.1s + 124.74 = 162
8.1s = 37.26
s = 4.6
Consider an urn containing four balls, numbered 110, 101, 011 and 000, from which one ball is drawn at random. For k = 1, 2, 3 let Ak be the event of drawing a ball with 1 in the kth position. Show your work in answering the following:a) Are A1, A2, and A3 pairwise independent?b) Are A1, A2, and A3 mutually independent?
a)The events A1, A2, and A3 are pairwise independent.
b) The events are not mutually independent.
What is probability?
Simply put, probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics is the study of events subject to probability.
Observe the given numbers:
Out of 4 numbers, 2 have 1 at first position
so outcomes in event A1 is
A1 ={ 110, 101}
Similarly, possible outcomes in A2 is
A2 ={110, 011}
A3 = {101,111}
The probabilities of events are
P(A1) = 2/4 = 0.5
P(A2) = 2/4 = 0.5
P(A3) = 2/4 = 0.5
Also,
(A1 and A2) = {110}
(A1 and A3) = {101}
(A2 and A3) = {011}
The probabilities are
P(A1 and A2) = 1 /4 = 0.25
P(A1 and A3) = 1 /4 = 0.25
P(A2 and A3) = 1 /4 = 0.25
a)
From above we have
P(A1 and A2) = P(A1)P(A2) = 0.25
P(A1 and A3) = P(A1)P(A3) = 0.25
P(A2 and A3) = P(A2)P(A3) = 0.25
So events A1, A2, and A3 are pairwise independent.
b)
Since P(A1 and A2), P(A1 and A3) and P(A2 and A3) are not equal to zero so events are not mutually independent.
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a)The events A1, A2, and A3 are pairwise independent.
b) The events are not mutually independent.
What is probability?
Simply put, probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics is the study of events subject to probability.
Observe the given numbers:
Out of 4 numbers, 2 have 1 at first position
so outcomes in event A1 is
A1 ={ 110, 101}
Similarly, possible outcomes in A2 is
A2 ={110, 011}
A3 = {101,111}
The probabilities of events are
P(A1) = 2/4 = 0.5
P(A2) = 2/4 = 0.5
P(A3) = 2/4 = 0.5
Also,
(A1 and A2) = {110}
(A1 and A3) = {101}
(A2 and A3) = {011}
The probabilities are
P(A1 and A2) = 1 /4 = 0.25
P(A1 and A3) = 1 /4 = 0.25
P(A2 and A3) = 1 /4 = 0.25
a)
From above we have
P(A1 and A2) = P(A1)P(A2) = 0.25
P(A1 and A3) = P(A1)P(A3) = 0.25
P(A2 and A3) = P(A2)P(A3) = 0.25
So events A1, A2, and A3 are pairwise independent.
b)
Since P(A1 and A2), P(A1 and A3) and P(A2 and A3) are not equal to zero so events are not mutually independent.
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Scott deposits $300 into an account that pays a simple interest at a rate of 4% per year. How much interest will he be paid in the first 6 years?
Answer:
$72
Step-by-step explanation:
300
4% per year
so 12$ per year,
6 years = 12*6 = 72
A flu epidemic occurs in Middletown schools and each student is either susceptible, sick, or infected. The percentage of students in each category by grade level is given below. Infected 70% 10% 20%
[S]= Sick 65% 15% 20%
Susceptible 60% 25% 15%
The population distribution of the Middletown school district is given below
HighSchool 1115 1100
[P] = MiddleSchool 830 795
Elementary 1610 1580
In this matrix question where percentages are listed, answers to all these questions are written below.
In the matrix given in the image, we can calculate following things:
a. S32 is a value that represents the percentage of sick students in the third grade of the elementary school. From data we can say it is 10%
b. P22 is a value that represents the number of middle school students in the second grade. From data we can say there are 795 such boys.
c. To find[tex][s] * [p][/tex] we need to multiply the percentage of students in each category by the corresponding population in each grade level.
[tex]\left[\begin{array}{ccc}1190.75&1311.75\\1194&1366.25\\1233&1405.5\end{array}\right][/tex]
Columns: Girls, Boys
Rows: Susceptible, Sick, Infected
d. [tex][[s] * [p]]21[/tex] represents the number of sick students in the second grade of the elementary school.
It represents that 1194 girls are sick.
e. To find the number of sick girls, we need to multiply the number of girls in each grade level by the corresponding percentage of sick students.
1194
f. To find the number of infected boys, we need to multiply the number of boys in each grade level by the corresponding percentage of infected students.
1406
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Complete Question: Image is uploaded for it.
a. Describe what S32 value is and what it will represent
b. Describe what P22 value is and what it will represent
c. Find [tex][s] * [p][/tex]
d. Describe what[tex][[s] * [p]]21[/tex] value is and what it will represent
e. How many sick girls are present?
f. Number of boys infected?
Select the correct word in each sentence :) Thanks!
A phase change occurs when the kinetic energy increases/decrease enough so that the attraction between molecules pulls them together.
A substance with weaker molecular attraction will freeze at a higher/lower temperature because it requires more/less energy to be TAKEN OUT before the attraction pulls the molecules together.
A substance with stronger molecular attraction will evaporate at a higher/lower temperature because it requires more/less energy to be ADDED to overcome attraction between the molecules.
A phase change occurs when the kinetic energy decrease enough so that the attraction between molecules pulls them together.
A substance with weaker molecular attraction will freeze at a lower temperature because it requires more energy to be taken out before the attraction pulls the molecules together.
A substance with stronger molecular attraction will evaporate at a higher temperature because it requires less energy to be added to overcome attraction between the molecules.
What is a phase change?In Science, a phase change refers to the change in the state of matter of a given chemical compound or substance. Additionally, some examples of a phase change include the following;
Solid to liquidLiquid to gasGas to solidLiquid to solidGenerally speaking, a change in the state (phase) of matter that results in the release of energy is known as an exothermic reaction and they include:
Deposition.Condensation.Freezing.In conclusion, sublimation, melting, and evaporation are other processes that are associated with a change in the state (phase) of matter.
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A city has cone-shaped buildings for storing salt and sand for icy roads. Each
building has a radius of 50 feet and a height of 30 feet. Find the volume of the
building. Use 3.14 for 7.
OA. 19,625 ft³
OB. 78,500 ft3
O C. 4710 ft³
OD. 235,500 ft3
What is the average atomic mass of the element in the data table?
37.96 is the average atomic mass of the element in the data table .
What does average mean?
The average is calculated by dividing the total number of values by the total number of values. It can also be described as mean. The mean in statistics is the average value across the specified sample or data set.
Mean, median, and mode are the three primary categories of average. Each of these methods operates a little bit differently and frequently yields values that diverge a little bit from the norm. The most popular type of average is the mean.
the average atomic mass = 35.97 + 37.96 + 39.96/3
= 113.89/3
= 37.96
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Answer:
39.96
Step-by-step explanation: I think
Simplify the following union and or insection
(1,13] ∩ [13,19)
The intersection in the below problem is 13
Given equation is
[1,13] ∩ [13,19]
We need to simplify the expression and tell the intersection of the expression
After doing the intersection, we will take all the common elements in two sets
so , we have
[1,13] ∩ [13,9 ] = 13
therefore we get the intersection of the given expression by simplifying it as 13.
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Let V be an inner product space, S and S0 be subsets of V, and W be a finite-dimensional subspace of V.
Prove:S0⊂S implies S⊥⊂S⊥0
Pf: Let x∈S⊥, for all y∈S, the inner product ⟨x,y⟩ is given ⟨x,y⟩=0
, which implies y∈S0 Then S0⊂S, x∈S⊥0.Proved.
Prove: W=(W⊥)⊥
Pf: Let x∈W and y∈W⊥. By definition, for all y∈W⊥, the inner product is given by ⟨x,y⟩=0
. Using x∈W, this gives W⊂(W⊥)⊥
How to prove the other side?
Let V be an inner product space, S and S0 be subsets of V, and W be a finite-dimensional subspace of V. To prove the other side,
1 ) Suppose (e1,…,en) is basis for W. Suppose x∈(W⊥)⊥. Consider the projection w of x onto W, specifically:
w=⟨x,e1⟩e1+…+⟨x,en⟩en∈W.
Then, x−w is perpendicular to each ei, since
⟨x−w,ei⟩=⟨x,ei⟩−∑j=1n⟨⟨x,ej⟩ej,ei⟩=⟨x,ei⟩−∑j=1n⟨x,ej⟩⟨ej,ei⟩=⟨x,ei⟩−∑j=1n⟨x,ej⟩δij=⟨x,ei⟩−⟨x,ei⟩=0.
Since the linear map ⟨⋅,x−w⟩ is constantly 0 on the basis of W, it is constantly zero on all of W, hence x−w∈W⊥.
But then, since x∈(W⊥)⊥, we have ⟨x−w,x⟩=0, and since ⟨x−w,w⟩=0,
∥x−w∥2=⟨x,x−w⟩−⟨w,x−w⟩=0,
i.e. x=w∈W, completing the proof.
2 ) Suppose that w∈(W⊥)⊥
then ⟨w,x⟩=0
for all x∈W⊥
If w∉W, then w=v+v′ where v∈W and v′∈W⊥∖{0}.
Then.
0=⟨w,x⟩=⟨v,x⟩+⟨v′,x⟩=⟨v′,x⟩
for all x∈W⊥.
However, v′∈W⊥ and ⟨v′,v′⟩≠0 so it is a contradiction.
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