use mathematical induction to prove 9. boole’s inequality, which states: for a sequence of events a1, a2, . . . , an

The probability that exactly four people get the right letter is 1/120.

Let the people form a line and the envelopes are on the table.

The first person has a 1 in 5 chance of selecting the right letter.  (He selects the correct one)

Now there are 4 on the table so the second person has a 1 in 4 chance of getting the right one (He selects the correct one)

Now there are 3 on the table so the second person has a 1 in 3 chance of getting the right one (He selects the correct one)

Now there are 2 on the table so the second person has a 1 in 2 chance of getting the right one (He selects the correct one)

and so on

so

P(everyone getting the correct letter) = 1/5*1/4*1/3*1/2*1/1=1/5!=1/120.

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Based on the definition of the inscribed angle of a semicircle , the value of x is: A. 6.

An inscribed angle of a semicircle is an angle whose vertex lies on the circumference of the semicircle and whose sides are chords of the semicircle. The measure of an inscribed angle of a semicircle is always equal to half the measure of the central angle (90 degrees) that intercepts the same arc as the inscribed angle.

Therefore, we would have:

16x - 6 = 90

16x = 90 + 6

16x = 96

16x/16 = 96/16

x = 6

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The variable represented in the statement "record the percentage change in the price of a stock for 100 stocks publicly traded on wall street." is a Quantitative Variable .

The Quantitative variables are defined as numerical which represent a measurement or a count.

In the statement :  the percentage change in the price of a stock is a numerical representation of the amount of change in the price of a stock, expressed as a ratio or a percentage.

Therefore ,  the variable "percentage change in the price of a stock" is a quantitative variable because it is numerical and represents a measurement of change in the price of a stock.

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The statement does not imply that there are any.

(a) i. ∃x (K(x) ∧ ¬K(x))

This statement translates to "there exists an x such that x is a knight and x is not a knight".

ii. We can conclude that there is at least one Knight and at least one Knave on the island, since the statement implies that there are some inhabitants who are both.

(b) i. ∀x ¬K(x)

This statement translates to "for all x, x is not a knight".

ii. We can conclude that there must be at least one Knave on the island, since the statement implies that all inhabitants are Knaves. However, we cannot conclude anything about the presence of Knights on the island, since the statement does not imply that there are any.

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The value of the following cardinalities:

a. |A| when A = {4, 5, 6,..., 37} is 34.

b. |A| when A = {x ∈ ℤ : -2 ≤ x ≤ 100} is 103.

c. |A∩B| when A = {x ∈ ℕ : x ≤ 20} and B = {x ∈ ℕ : x is prime} is 8.

We have to determine the following cardinalities.

a. |A| when A = {4, 5, 6,..., 37}.

Total number of elements = 34

|A| = 34

b. |A| when A = {x ∈ ℤ : -2 ≤ x ≤ 100}.

|A|: The number of integer that lies between -2 ≤ x ≤ 100

So the total number of elements from -2 ≤ x ≤ 100 is 103.

So, |A| = 103

c. |A∩B| when A = {x ∈ ℕ : x ≤ 20} and B = {x ∈ ℕ : x is prime}.

A∩B = {x ∈ ℕ : x ≤ 20 and x is prime}

A∩B = {2, 3, 5, 7, 11, 13, 17, 19}

Total elements = 8

|A∩B| = 8

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1. The solution of a system of 3 simultaneous linear equations in three variables are (2, 3, 1).

2. The base of the mathematical constant e, an irrational and transcendental number roughly equal to 2.718281828459, is the natural logarithm of a number.

1. A system of 3 simultaneous linear equations in three variables are

5x - 4y + 2z = 0.............(1)

-3x + 4y = 6.............(2)

x + 4z = 6.............(3)

We solve these system of equations using the substitution method and elimination method.

Add equation (1) and (2)

5x - 4y + 2z + (-3x + 4y) = 0 + 6

5x - 4y + 2z - 3x + 4y = 0 + 6

After the elimination the equation is:

2x + 2z = 6

Taking common 2 and divide by 2 on both side, we get

x + z = 3................(4)

Subtract equation 3 and 4

x + 4z - (x + z) = 6 - 3

x + 4z - x - z = 3

3z = 3

Divide by 3 on both side, we get

z = 1

Now putting the value of z in equation 4

x + 1 = 3

Subtract 1 on both side, we get

x = 2

Now putting the value of x in equation 2

-3(2) + 4y = 6

-6 + 4y = 6

Add 6 on both side, we get

4y = 12

Divide by 4 on both side, we get

y = 3

Now the solution of system of equation is (2, 3, 1).

2. The exponent or power that must be added to a base to produce a particular number is called the logarithm. If bx = n, then the mathematical expression for x is x = logb n, where x is the logarithm of n to the base b.

With infinite decimal places and no discernible pattern, the natural base e is an irrational number. The initials of the number e are 2.718281828459.

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The complete question is:

1. Solve a system of 3 simultaneous linear equations in three variables.

5x - 4y + 2z = 0

-3x + 4y = 6

x + 4z = 6

2. What is a logarithm? How is e, the natural logarithm base, defined?

0.079 liters is 79.1 ml with correct significant numbers.

We know that 1 liters = 1000 ml

Hence, 1 ml = 0.001 liters

In the given question we need to convert 79.1 ml to liters. So, we multiply 79.1 ml by 0.001 l to convert ml to liters.

79.1 x 0.001 = 0.0791 l

The given question 79.1 ml has 3 significant figures. Hence, we can write 0.0791 as 0.079 l as 3 significant figures, counting the digits on the right of decimal point.

Therefore, 0.0791 liters is 79.1 ml. On rounding it to correct significant numbers, we get 0.079 l.

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Deductive reasoning is used.

Deductive reasoning is the logic process of drawing a conclusion applying a general rule to a specific case.

To solve the equation 5x + 7 = 12 you apply this:

Rule                                                        application      

Subtraction property of equality          5x = 12 - 7

Simplification                                         5x = 5

Division property of equality                 x = 5/5

Simplification                                          x = 1.

Thus, by using the general rules to the specific equation, you get the result x = 1, which shows that this is a case of deductive reasoning.

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Demetri lives in the city and sometimes uses a ride share service. A ride costs $1.50 for the first 1/5 mile and $0.25 for each additional 1/5 mile. Demetri does not want to spend more than $3.75 on a ride. So, he must not go beyond 9 miles.

An expression that contains variables, constants, and algebraic operations is defined as just an algebraic expression in mathematics (addition, subtraction, etc.). Terms comprise expressions. A mathematical statement with variables, constants, coefficients, and algebraic operations is known as an algebraic expression.  Terms comprise expressions. Algebraic equations are declarations of the equality of two expressions created by performing the algebraic operations of adding, subtracting, multiplying, dividing, raising to a power, and root extraction to a collection of variables.

Given data

A ride costs $1.50 for the first 1/5 mile.

$0.25 for each additional mile.

Let the number of miles be x

and the cost of x miles be y

Hence the expression for the total cost is

y = 1.5+ 0.25x

now when y = 3.75

or, 3.75 = 1.5 + 0.25x

or, 3.75 - 1.5 = 0.25x

or, 2.25 = 0.25x

or, x = 2.25/0.25

or, x = 9 miles

Hence he must not go beyond 9 miles.

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The expression of the composite function is (d) 192 - 24t

From the question, we have the following parameters that can be used in our computation:

B(t) = 2(8 - t)

M(n) = 12n

To calculate the required function, we make use of

M(B(t))

Substitute the known values in the above equation, so, we have the following representation

M(t) = 12 * 2(8 - t)

So, we have

B(t) = 192 - 24t

Hence, the expression is (d) 192 - 24t

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The area in which the goat can haze inside the land with the shape of a triangle ABC = 16m²

The formula that is used to calculate the area of a triangle = 1/2 base ×height.

where base of a triangle = 4m

The height of the triangle = 8m

The area of the triangle = 1/2× 4× 8

= 16m².

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Complete question:

An unfenced grassland is in the form of a right triangle ABC with AB=4m, BC=8m, and AC as the hypotenuse.if the land was fenced on sides ab and bc, determine the area in which the goat can graze inside the land abc.

The area in which the goat can haze inside the land ABC is; 16m²

The formula that is usually applied to calculate the area of a triangle is expressed as;

Area =  ¹/₂ * base * height.

The parameters are;

Base of a triangle = 4m

Height of the triangle = 8m

Thus,

Area of the triangle ABC = ¹/₂ * 4 * 8

Area =  16m².

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Complete question is:

An unfenced grassland is in the form of a right triangle ABC with AB=4m, BC=8m, and AC as the hypotenuse. If the land was fenced on sides ab and bc, determine the area in which the goat can graze inside the land abc.

Answer:

1.5

Step-by-step explanation:

42/28 =6/4=1.5

33/1.5=22

Insertion Sort with Binary Search is still a linear time-complexity algorithm in the average and worst case, but it is not the best option for sorting large arrays of elements, as the time complexity remains quadratic.

Insertion Sort is a simple sorting algorithm that works by inserting each element in its correct position in a sorted list.

a. It uses Θ(n²) comparisons in the worst case: This statement is true. In the worst-case scenario, Insertion Sort with Binary Search still takes Θ(n²) comparisons to sort an array of n elements. This happens when the array is sorted in descending order, and the elements need to be inserted one by one at the beginning of the list.

b. It uses Θ(n log n) comparisons in the average case: This statement is false. Insertion Sort with Binary Search still uses Θ(n²) comparisons in the average case, even though the use of Binary Search reduces the number of comparisons compared to Linear Search.

c. It uses Θ(n) comparisons in the best case: This statement is false. In the best-case scenario, Insertion Sort with Binary Search still takes Θ(n²) comparisons, as every element needs to be compared and inserted in its correct position

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Answer:

The function table for g(x) = -x + 4 is as follows:

X Y -5 9 -1 5 0 4 1 3 2 2

Answer:

9 m , 23 m , 23 m

Step-by-step explanation:

let the third side be represented by x , then the 2 equal sides are

3x - 4 and 3x - 4

the perimeter (P) is the sum of the 3 sides

P = x + 3x - 4 + 3x - 4 = 7x - 8

given P = 55 m , then equating

7x - 8 = 55 ( add 8 to both sides )

7x = 63 ( divide both sides by 7 )

x = 9

3x - 4 = 3(9) - 4 = 27 - 4 = 23

sides of triangle are 9 m , 23 m and 23 m

The equation of the secant line containing the points (-7,g(-7)) and (8,g(8)) is

y = (123/15)x + (98 - (123/15) * (-7))

The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept. To find the equation of the secant line, we need to find the slope (m) and the point of intersection (x,y) of the line with the y-axis.

To find the slope (m) of the secant line, we can use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1,y1) = (-7, g(-7)) and (x2,y2) = (8, g(8)).

So, m = (g(8) - g(-7)) / (8 - (-7)) = (g(8) - g(-7)) / 15

Now, we can substitute the value of g(x) in the above equation and get the value of m

g(x) = 2x^2-7

m = (2(8^2)-7 - 2(-7^2)-7) / 15 = (2(64)-7 - 2(49)-7) / 15 = (128-7-98-7) / 15 = 123/15

To find the point of intersection (x,y) of the line with the y-axis, we can use one of the given points and the slope. Let's use the point (-7,g(-7))

y = mx + b

g(-7) = m * (-7) + b

b = g(-7) - m * (-7)

We can substitute the value of g(x), m and x = -7 in the above equation and get the value of b.

g(x) = 2x^2-7

b = 2(-7)^2-7 - m * (-7) = 2(49)-7 - 123/15 * (-7) = 98 - 123/15 * (-7)

So, the equation of the secant line containing the points (-7,g(-7)) and (8,g(8)) is

y = (123/15)x + (98 - (123/15) * (-7))

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Answer:

Step-by-step explanation:

2

Area of circle A = pi r²

r is the radius

small circle area = pi

large circle area = k.pi

the diameter of the the large circle equals 3 times the diameter of the small circle .

since A = pi . r²

pi.r² = pi as given the area of the small circle

r²= 1

r= 1 is the radius of the small circle the the diameter = 2

the diameter of the large circle = 3.2 = 6

the radius = 3

the area = pi .3² = 9 pi

then k = 9

The columns of a are linearly dependent, we can conclude that y = (4, 9) is not in the range of t(x).

What is vector ?

A vector is a mathematical object that has both magnitude (or length) and direction. Vectors can be used to represent physical quantities such as velocity, force, or displacement, where the magnitude represents the size of the quantity and the direction represents its orientation in space.

The range of t(x) = ax, where a = 1 −3 0 5 3 1, is the set of all possible values of t(x), which is the set of all linear combinations of the columns of the matrix a. To check if y = (4, 9) is in the range of t(x), we need to see if there exists a vector x such that t(x) = y. If a vector x exists, then y = (4, 9) is in the range of t(x). If no such vector exists, then y is not in the range of t(x).

Since the columns of a are linearly dependent, we can conclude that y = (4, 9) is not in the range of t(x).

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The function h(x) can be expressed as a composition of two other functions: f(x) and g(x).

In mathematics, functions are a way to describe relationships between variables.

The function f(x) is defined as 2 raised to the power of x. This means that for any value of x, the output of the function f(x) will be 2 raised to that power.

If x is 3, then f(3) = 2³ = 8.

The function g(x) is defined as x. This means that the output of the function g(x) will be the same as its input.

If x is 4, then g(4) = 4.

Combining these two functions, we can express h(x) as f(g(x) + 6). This means that for any value of x, the output of h(x) will be 2 raised to the power of g(x) plus 6.

If x is 2, then g(2) = 2, and

=> h(2) = f(g(2) + 6) = f(8) = 2⁸ = 256.

In summary, the function f(x) raises 2 to the power of x, and the function g(x) returns its input.

By combining these two functions, we can express h(x) as the result of 2 raised to the power of g(x) plus 6.

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Answer:

By using what we know about right triangles, we will see that the hobbits are at 744.39 meters from the Dark Tower.

This position can be visualized as a right triangle, with the catheti being the tower's height and the distance between the hobbits and the Dark Tower.

Then we can use the relation:

Tan(θ) = (opposite cathetus)/(adjacent cathetus)

Where θ = 28°

The adjacent cathetus to this angle exists the height of the tower, then:

Adjacent cathetus = 1,400 m

The opposite cathetus exists the distance,

Tan(28°) = D/1,400m

Solving for D we get:

Tan(28°)*1,400m = D = 744.39m

The hobbits are at 744.39 meters from the Dark Tower.

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since 4000 ---> 210

then x -----> 336

(4000 * 336) / 210 = x

x = 6,400

Answer:

5.8

Step-by-step explanation:

[tex]\tan 58^{\circ}=\frac{x}{3.6} \\ \\ x=3.6\tan 58^{\circ} \\ \\ x \approx 5.8[/tex]

The solution to the inequality -3 < 2x - 3 is given as follows:

x > 0.

The inequality for this problem is defined as follows:

-3 < 2x - 3.

It can be rewritten as follows:

2x - 3 > -3.

We isolate the variable x, similarly to an equality, as follows:

2x > 0

x > 0/2

x > 0.

Hence all the options that have positive numbers are solutions to the inequality−3<2x−3.

The problem is incomplete, hence the answer was given on general terms.

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No, lecture notes collected during a bachelor degree study are not a type of date. Lecture notes are a collection of the notes taken by a student during the lectures in the course of their bachelor degree study.

They are not the same as dates, which are the specified days on which something happens or is planned to happen. Lecture notes are usually written in the form of summaries of a lecture, which are usually made up of the important points that the student has noted down during the lecture.

They may also include diagrams, equations, and other visual aids that the student has used to illustrate the material covered. Lecture notes are not the same as dates, as they are not related to any specified days. They are simply summaries of the lecture topics discussed by the professor in the course of the student’s bachelor degree study.

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Answer:

Below

Step-by-step explanation:

Typo in your post ?

.875   IS   rational    it equals 7/8

(-1, 1) is the required ordered pair that represents a solution to both equations. Option C is correct.

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

Here,
On the given graph there are two lines intersecting at points (-1, 1), since the point of intersection of the two lines represents the solution to both equations.

Thus, (-1, 1) is the required ordered pair that represents a solution to both equations.

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Answer:

(-1, 1) is the required ordered pair that represents a solution to both equations. Option C is correct.

What are simultaneous linear equations?

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

Here,

On the given graph there are two lines intersecting at points (-1, 1), since the point of intersection of the two lines represents the solution to both equations.

Thus, (-1, 1) is the required ordered pair that represents a solution to both equations.

Step-by-step explanation:

In rapidminer, correlation matrix is used to create correlation coefficients.

Correlation matrix Operator determines correlation between all attributes and it can produce a weights vector based on these correlations. Correlation is a statistical technique that can show whether and how strongly pairs of attributes are related.

Suppose we have two Attributes X and Y, with means X' and Y' respectively and standard deviations S(X) and S(Y) respectively. The correlation is computed as summation from 1 to n of the product (X(i)-X').(Y(i)-Y') and then dividing this summation by the product (n-1).S(X).S(Y) where n is total number of Examples and i is the increment variable of summation.

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Answer:

Step-by-step explanation:

18+4(m-3)+4m

18+4m-12+4m

=8m+6

Answer:  options a and c