Show that {(x, y) | x − y ∈ Q} is an equivalence relation on the set of real numbers, where Q denotes the set of rational numbers. What are [1], [1/2], and [π]?

The length of HJ is approximately 7.754 units and angle G is approximately 61.55 degrees.

To find the remaining side and angles of the right-angled triangle GHJ, we can use the Pythagorean theorem and trigonometric functions.

Using the Pythagorean theorem, we know that in a right-angled triangle:

[tex]HJ^2 + GH^2 = GJ^2[/tex]

Substituting the given values:

[tex]HJ^2 + 14^2 = 16^2\\\\HJ^2 + 196 = 256\\\\HJ^2 = 256 - 196\\\\HJ^2 = 60[/tex]

Taking the square root of both sides:

[tex]HJ = \sqrt{(60)}\\\\HJ = 7.754[/tex]

To find angle G, we can use the sine function:

sin(G) = GH / GJ

sin(G) = 14 / 16

sin(G) = 0.875

G ≈ arcsin(0.875)

G ≈ 61.55 degrees

Therefore, angle G is approximately 61.55 degrees.

Thus from the properties of a triangle,

∠G+∠H+∠J=180°

61.55+90+∠J=180°

∠J = 90-61.55

∠J = 28.45°

Therefore, the measurement of angle J is 28.45°.

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Answer:

Option: D

Step-by-step explanation:

To find the quotient of −219 and 3, we perform the division:

-73

_________

3 | -219

- 21

-----

9

The quotient of −219 and 3 is -73. Therefore, the correct answer is D: -73.

The number of hot water bottles sold is 12.

Let the number of hot water bottles sold be x.

Given that the customer bought 10 times as many thermometers as hot water bottles.

So the number of number of thermometer is = 10x.

Cost of each thermometers is = $2.

So the cost of '10x' number of thermometers is = $2*10x = $20x.

Cost of each hot water bottles = $6.

So the cost of 'x' hot water bottles is = $6x.

So the total sales = $20x + $6x = $26x.

Given that the total sales according to data is $312.

So the equation which best fit this situation is,

26x = 312

x = 312/26 = 12

Hence the number of hot water bottles sold is 12.

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=======================================================

Explanation:

Let h be the height of largest triangle. It is the vertical segment in the diagram.

The three triangles are similar, so we can form the proportion shown below to solve for h.

7/h = h/35

7*35 = h*h

245 = h^2

h = sqrt(245)

------------------

Now focus on the medium-sized triangle on the right. It has legs of 35 and sqrt(245). The hypotenuse is x.

Use the pythagorean theorem.

a^2 + b^2 = c^2

(sqrt(245))^2 + (35)^2 = x^2

245 + 1225 = x^2

x^2 = 1470

x = sqrt(1470)

x = sqrt(49*30)

x = sqrt(49)*sqrt(30)

x = 7*sqrt(30)

According to the question we have , the opportunity cost for Costa Rica to produce 1 ton of coffee is 1 ton of coconuts.

To determine the opportunity cost for Costa Rica to produce 1 ton of coffee, we need to look at the production possibilities of both goods in Jamaica and Costa Rica. From the table, we can see that Jamaica can produce 1 ton of coconuts or 1 ton of coffee, while Costa Rica can produce 1 ton of coconuts or 2 tons of coffee.

Therefore, the opportunity cost for Costa Rica to produce 1 ton of coffee is the amount of coconuts that Costa Rica must give up to produce 1 ton of coffee. In this case, Costa Rica could produce 2 tons of coffee or 1 ton of coconuts. Therefore, the opportunity cost for Costa Rica to produce 1 ton of coffee is 1 ton of coconuts.

In other words, if Costa Rica decides to produce 1 ton of coffee, it will have to give up the production of 1 ton of coconuts. This is the opportunity cost of producing coffee instead of coconuts.

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For the given correlations, the data points would be most widely scattered around the regression line when the correlation coefficient (r) is closest to 0. In this case, the correct option is a. r = 0.10. As the correlation coefficient approaches 1 or -1, the data points become more closely clustered around the regression line.

Correlation refers to the strength and direction of the relationship between two variables. A correlation coefficient (r) ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation between the two variables.

Now, coming to the scatter of data points around the regression line, it is measured by the standard error of estimate (SEE). SEE represents the average amount by which the observed values differ from the predicted values. The larger the SEE, the more scattered the data points are around the regression line.

Therefore, the answer to the given question would be option (a) r = 0.10. This is because a correlation coefficient of 0.10 indicates a weak positive relationship between the variables, and the data points would be more widely scattered around the regression line, resulting in a larger SEE.

In summary, the scatter of data points around the regression line is inversely related to the strength of correlation, and the data points would be most widely scattered around the regression line for a weak correlation (r = 0.10).

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Check the picture below.

The peak P is 8,687 feet above sea level.

Let's denote the height of peak P as h.

Using the information given, we can set up two right triangles: △QAP and △QBD.

In △QAP, we have the opposite side (AP) and adjacent side (AQ). Since we know angle QAP is 20°, we can use the tangent function:

tan(20°) = AP / AQ

We wlll rearrange to solve for AP:

AP = AQ * tan(20°)

In △QBD, we have the opposite side (BD) and the adjacent side (DQ). Since we know angle QBD is 35°, we will use the tangent function:

tan(35°) = BD / DQ

We will rearrange to solve for BD:

BD = DQ * tan(35°)

Now, let's calculate the values:

AP = 6500 ft * tan(20°) = 2323.84 ft

BD = 230 ft * tan(35°) = 136.41 ft

Difference in elevation :

=  2323.84 ft - 136.41 ft

= 2187.43 ft

The elevation at peak P:

= 6500 ft + 2187.43 ft

= 8687.43 ft.

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This statement is true, if 2 is an eigenvalue of A, A - 21 is not invertible.

If 2 is an eigenvalue of matrix A, then A - 21 is indeed not invertible. An eigenvalue represents a scalar value that characterizes certain properties of a matrix. The eigenvalues of a matrix A can be obtained by solving the characteristic equation |A - λI| = 0, where λ is the eigenvalue and I is the identity matrix. In this case, if 2 is an eigenvalue of A, it means that A - 2I has a determinant of 0. Therefore, if we subtract 21 from A, we get (A - 21) - 2I, which still has a determinant of 0. Consequently, A - 21 is not invertible.

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The sum of all the numbers on the chessboard is 4036, obtained by summing the values in the top row, leftmost column, and the products in the remaining squares.

The sum of all the numbers on the chessboard can be found by summing the values in the top row, the values in the leftmost column, and the products in the remaining squares.

The sum of the values in the top row is 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255.

The sum of the values in the leftmost column is 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 = 3270.

The sum of the products in the remaining squares can be calculated by finding the sum of the geometric series 1 + 2 + 4 + ..... + 128. Using the formula for the sum of a geometric series, this sum is 2⁸⁺¹ - 1 = 2⁹ - 1 = 511.

Therefore, the sum of all the numbers on the chessboard is 255 + 3270 + 511 = 4036.

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The integral of [tex]x^3/\sqrt(x^2+4)[/tex] using trigonometric substitution is: [tex]8 * (1/3)tan^2\theta(1 + tan^2\theta)^(3/2) + C[/tex], where θ is determined by x = 2tanθ, and C represents the constant of integration

What is integration?

Integration is a fundamental concept in calculus that involves finding the integral of a function.

To integrate the function [tex]\int(x^3/\sqrt(x^2+4))[/tex] dx using a trigonometric substitution, we can use the substitution x = 2tanθ. Let's go through the steps:

Substitute x = 2tanθ. This implies [tex]dx = 2sec^2\theta d\theta.[/tex]

Rewrite the integral in terms of θ:

[tex]\int((8tan^3\theta)/(\sqrt(4tan^2\theta+4))) * 2sec^2\theta d\theta.[/tex]

Simplify the expression inside the square root:

[tex]\int((8tan^3\theta)/(2sec\theta)) * 2sec^2\theta d\theta.\\\\\int(8tan^3\theta) * sec\theta d\theta.[/tex]

Simplify further:

[tex]16\int tan^3\theta sec\theta d\theta.[/tex]

Apply the trigonometric identity: [tex]sec^2\theta = 1 + tan^2\theta[/tex]. Rearranging, we get: [tex]sec\theta = \sqrt(1 + tan^2\theta).[/tex]

Substitute [tex]sec\theta = \sqrt(1 + tan^2\theta)[/tex] in the integral:

[tex]16\int tan^3\theta * \sqrt(1 + tan^2\theta) d\theta.[/tex]

Let u = tanθ, which implies [tex]du = sec^2\theta d\theta[/tex]. We can rewrite the integral in terms of u:

[tex]16\int u^3 * \sqrt(1 + u^2) du.[/tex]

Now we have a rational power of u. We can use the substitution [tex]v = 1 + u^2[/tex] to simplify it:

[tex]v = 1 + u^2[/tex], which implies dv = 2u du.

Rewrite the integral using v:

[tex]16\int (u^3 * \sqrt v) * (1/2u) dv.\\\\8\int (u^2\sqrt v) dv.[/tex]

Simplify and integrate:

[tex]8\int (u^2\sqrt v) dv = 8\int(u^2 * v^{(1/2)}) dv = 8\int u^2v^{(1/2)} dv.[/tex]

Integrate [tex]u^2v^{(1/2)[/tex] with respect to v:

[tex]8 * (1/3)u^2v^{(3/2)} + C.[/tex]

Replace v with [tex]1 + u^2[/tex]:

[tex]8 * (1/3)u^2(1 + u^2)^{(3/2)} + C.[/tex]

Substitute u = tanθ back into the expression:

[tex]8 * (1/3)tan^2\theta(1 + tan^2\theta)^{(3/2)} + C.[/tex]

So, the integral of [tex]x^3/\sqrt(x^2+4)[/tex] using trigonometric substitution is:

[tex]8 * (1/3)tan^2\theta(1 + tan^2\theta)^{(3/2)} + C,[/tex]

where θ is determined by x = 2tanθ, and C represents the constant of integration

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True. Since the square matrix that represents the DCT coefficient is an orthogonal matrix, its inverse and transpose are the same. This property is a fundamental characteristic of orthogonal matrices and holds true for all orthogonal matrices.

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False While it is true that the square matrix that represents the DCT coefficient is orthogonal, it does not mean that its inverse and transpose are the same.

the inverse of an orthogonal matrix is its transpose. However, the transpose of the matrix does not necessarily mean that it is the same as the inverse of the matrix. In the statement is false because the inverse and transpose of an orthogonal matrix are not always the same.


The Discrete Cosine Transform (DCT) coefficient matrix is an orthogonal matrix. In the case of orthogonal matrices, the inverse matrix is indeed equal to the transpose of the original matrix. This is because the product of an orthogonal matrix and its transpose results in the identity matrix.

Since the square matrix representing the DCT coefficient is an orthogonal matrix, its inverse and transpose are the same.

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To find the centroid for the quarter circle using polar coordinates, we need to first determine the equations for the centroid coordinates (r,θ).

The formula for the centroid coordinates are:
r = (1/A) ∫∫ r^2 sin(θ) dr dθ
θ = (1/A) ∫∫ 0.5r^2 cos(θ) dr dθ
where A is the area of the lamina, which is given as π/4 in this problem.
For the quarter circle, we know that the radius (r) ranges from 0 to R (where R is the radius of the circle) and the angle (θ) ranges from 0 to π/2. Therefore, we can rewrite the integrals as follows:
r = (1/π/4) ∫0^(π/2) ∫0^R r^2 sin(θ) dr dθ
θ = (1/π/4) ∫0^(π/2) ∫0^R 0.5r^2 cos(θ) dr dθ
Simplifying these integrals and solving for r and θ, we get:
r = (4/πR^4) ∫0^(π/2) (1-cos^3(θ)) dθ
θ = (2/πR^4) ∫0^(π/2) (1-cos^2(θ)) sin(θ) dθ
Evaluating these integrals, we get:
r = (8R/9π)
θ = (4R/3π)
Therefore, the centroid coordinates (r,θ) for the quarter circle are:
r = (8R/9π)
θ = (4R/3π)
Note that these coordinates are relative to the origin of the polar coordinate system. To find the absolute centroid coordinates, we need to add the coordinates of the center of the circle to these values.

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The tables that can be used to verify this are the original function table and the inverse function table.

To verify that two functions are inverses of each other, we need to check if applying one function followed by the other gives the original input value. Therefore, the tables that can be used to verify this are the original function table and the inverse function table.

Let's assume the original function is represented by Table A, which maps inputs to outputs, and the inverse function is represented by Table B, which maps outputs back to inputs.

To verify the functions' inverses, we need to compare the values of Table A and Table B. If for every input in Table A, its corresponding output in Table B matches the original input value, and vice versa, we can conclude that the functions are inverses of each other.

By comparing the two tables, we can determine if there is a consistent reversal of inputs and outputs, confirming the inverse relationship between the functions.

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Large mechanoreceptor axons from the face region cross over contralaterally at the level of the trigeminal nerve, which is the fifth cranial nerve and is responsible for sensation in the face and motor functions such as chewing. The correct answer is b) Trigeminal nerve.

The other options, spinal cord nerves, dorsal column, and nasal cavity, are not directly involved in the crossover of these axons. Large mechanoreceptor axons from the face region cross over contralaterally at the level of the trigeminal nerve. The trigeminal nerve is responsible for sensory information from the face, including touch, temperature, and pain.

It consists of three main branches: the ophthalmic nerve (V1), the maxillary nerve (V2), and the mandibular nerve (V3). The trigeminal nerve carries the sensory information from the face to the brain, and the crossover of axons occurs at the trigeminal nerve level before the information is transmitted further. Hence, b is the correct option.

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The operationalization of the used constructs of the model is not typically considered a part of the theoretical aspects of a model. The correct option is (B).

Operationalization refers to the process of defining and measuring the variables or constructs in a study. It involves turning abstract concepts into observable and measurable variables.

On the other hand, the theoretical parts of a model typically include:

A. The hypotheses corresponding to the model: This refers to the specific statements or predictions derived from the theoretical framework of the model.

Hypotheses provide testable expectations about the relationships between variables in the model.

C. A logical explanation of the relationship within a model: This involves providing a theoretical rationale or logical explanation for the expected relationships between variables in the model.

It is the theoretical framework that underlies the model and helps in understanding the underlying mechanisms or processes.

D. A graphical representation of the model: Graphical representation, such as diagrams or visual models, can be used to depict the structure or relationships within a model. It helps in visualizing the connections between variables and understanding the overall framework.

Therefore, B. The operationalization of the used constructs of the model does not belong to the theoretical parts of a model.

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$2000\\ r=rate\to 6.2\%\to \frac{6.2}{100}\dotfill &0.062\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &5 \end{cases}[/tex]

[tex]A = 2000\left(1+\frac{0.062}{1}\right)^{1\cdot 5}\implies A=2000(1.062)^5 \implies A \approx 2701.80 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{earned interest}}{2701.80~~ - ~~2000} ~~ \approx ~~ \text{\LARGE 701.80}[/tex]

Answer:

231.77

we have length as9.8cm

breadth as 5.5cm

height as 4.3cm

solution

we know that

volume =l*b*h

=9.8*5.5*4.3

=231.77

To find the unknown values in a right-angled triangle, such as the length of one side (a), we need additional information. Without specific measurements or relationships provided, the value of a cannot be determined.

In a right-angled triangle, the side lengths are typically denoted as a, b, and c, where a and b are the lengths of the two shorter sides (legs) and c is the length of the hypotenuse. To find the length of side a, we typically need either the lengths of the other sides (b and c), or additional information such as angles or ratios involving the sides.

The calculation of side lengths in a right-angled triangle often relies on trigonometric functions like sine, cosine, or tangent, or the Pythagorean theorem. These relationships allow us to solve for unknown side lengths when given certain measurements or angles.

Without specific values for side b or any additional relationships or measurements provided, it is not possible to determine the length of side a in the right-angled triangle. The determination of side lengths in a right-angled triangle requires further information or the application of relevant formulas or relationships associated with right triangles.

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The total area under the standard normal is 0.0760.

What is the standard normal?

A normal distribution with a mean of zero and a standard deviation of one is known as the standard normal distribution. The standard normal distribution is centered at zero, and the standard deviation indicates how much a specific measurement deviates from the mean.

Here, we have

Given: the total area under the standard normal curve to the left of z = -1.48 or to the right of z = 2.48.

= P(z < -1.48 or z > 2.48)

= P(z < -1.48) + P(z > 2.48)

= 0.0694 + 0.0066

= 0.0760

Hence, the total area under the standard normal is 0.0760.

Question: Determine the total area under the standard normal curve to the left of z= -1.48 or to the right of z = 2.48.

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Answer:

Area of a cricle:  [tex]A = \pi r^2[/tex]

Subject of the formula for radius is:

[tex]r= \sqrt{ \frac{A}{\pi }}[/tex]

Now replacing these values for circle 1, we get:

[tex]r= \sqrt{ \frac{490.63}{\pi }} = 12.5ft[/tex]

Similar for circle 2:

[tex]r= \sqrt{ \frac{219.45}{\pi }} = 8.36 in[/tex]

Answer:

please see detailed answers below

Step-by-step explanation:

Area of circle = π r ²

Circumference = π X D (D = diameter = 2 X radius)

1st circle:

Area of circle = π r ²

490.63 = π r ²

r ² = 490.63/π

r = ± √( 490.63/π )  ...... we only need the + (positive result since this is a length).

r = 12.5 feet. Diameter = 2r = 2(12.5) = 25 feet.

2nd circle:

Area of circle = π r ²

219.45 =  π r ²

r² = 219.45/π

r = √(219.45/π )

= 8.36 inches. diameter = 2r = 16.7 inches.

The given statement, "Block designs result only from observing subjects several times, each time with a different treatment," is false because block designs can be created by observing subjects several times with the same treatment as well.

Block designs involve dividing subjects into homogeneous groups or blocks based on certain characteristics, and within each block, subjects receive different treatments.

By doing so, the effects of confounding variables can be minimized, allowing for more accurate analysis of treatment effects. Thus, block designs can be used to compare treatments within subjects observed multiple times, whether the treatments are the same or different.

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Answer:

yw;)

Step-by-step explanation:

First, let’s find the derivatives of x1(t) and x2(t):

x1'(t) = d/dt [2e^(-t) - 2e^(-2t)] = -2e^(-t) + 4e^(-2t)

x2'(t) = d/dt [e^(-t) - 2e^(-2t)] = -e^(-t) + 4e^(-2t)

Now let’s substitute these derivatives into the given system of equations:

x1' = -2x2 becomes -2e^(-t) + 4e^(-2t) = -2(e^(-t) - 2e^(-2t)), which simplifies to -2e^(-t) + 4e^(-2t) = -2e^(-t) + 4e^(-2t).

x2' = x1 - 3x2 becomes -e^(-t) + 4e^(-2t) = (2e^(-t) - 2e^(-2t)) - 3(e^(-t) - 2e^(-2t)), which simplifies to -e^(-t) + 4e^(-2t) = -e^(-t) + 4e^(-2t).

Since both equations are true, we can conclude that x1(t) and x2(t) do indeed solve the given system of equations.

The probability that all three wires will meet the specifications is approximately 0.173 .

Expected value (mean) of wire resistance = 0.13 ohms Standard deviation of wire resistance = 0.005 ohms

the probability for each wire, we need to standardize the range of resistance values using the expected value and standard deviation. We can use the Z-score formula:

Z = (X - μ) / σ

Z is the standard score (Z-score) X is the observed value (resistance) μ is the mean (expected value) σ is the standard deviation

For the lower specification of 0.10 ohms

Z1 = (0.10 - 0.13) / 0.005

For the upper specification of 0.17 ohms

Z2 = (0.17 - 0.13) / 0.005

Using a standard normal distribution table , we can find the probability associated with each Z-score.

Lower bound of standardized range = (0.10 - 0.13) / 0.005 = -0.06

Upper bound of standardized range = (0.17 - 0.13) / 0.005 = 0.80

Let's calculate the probabilities for each wire

P(z < -0.60) ≈ 0.2743

P(z < 0.80) ≈ 0.7881

Since we want the probability that all three wires meet the specifications, we need to multiply these probabilities together since the wires are selected independently.

P(all three wires meet specifications) = P(z < -0.60) × P(z < 0.80) × P(z < 0.80)

P(all three wires meet specifications) ≈ 0.2743 × 0.7881 × 0.7881 ≈ 0.1703

Therefore, the probability that all three wires will meet the specifications is approximately 0.173, or 17.3% .

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Answer:120

Step-by-step explanation:

The orthogonal complement of s is spanned by [0, -1, 0].

The direct sum s ⊕ s includes all combinations of vectors from s and its orthogonal complement.

The orthogonal complement of a subspace s is the set of all vectors that are perpendicular to every vector in s. In this case, the subspace s is spanned by the vectors [0, 1, 0] and [9, 0, 1].

To find the orthogonal complement, we need to find the vectors that are perpendicular to both of these vectors.

Let's call the orthogonal complement of s as s⊥. To find s⊥, we need to find vectors [a, b, c] such that the dot product of [a, b, c] with both [0, 1, 0] and [9, 0, 1] is zero. We can set up two equations:

Solving these equations, we find that any vector of the form [0, -c, 0] is perpendicular to s. Therefore, the orthogonal complement s⊥ is the subspace spanned by the vector [0, -1, 0].

The direct sum s ⊕ s is the set of all vectors that can be expressed as a sum of vectors from s and s⊥. In this case, s ⊕ s is the set of all vectors that can be expressed as a sum of the vectors [0, 1, 0] and [9, 0, 1] along with the vector [0, -1, 0]. It represents the combination of all possible linear combinations of vectors from both s and s⊥.

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If the value of the sample covariance between the two random variables x and y equals -150, then we can conclude that x and y have a (an)

negative covariance.

Covariance is a measure of the relationship between two random variables. It indicates the extent to which the variables vary together.

A positive covariance suggests that the variables tend to move in the same direction, while a negative covariance indicates that the variables tend to move in opposite directions.

It is calculated by taking the average of the product of the differences between the values of X and their mean, and the values of Y and their mean.

In this case, since the sample covariance is -150, it implies that as the values of variable x increase, the values of variable y tend to decrease, and vice versa.

Thus, x and y have a negative relationship or negative covariance.

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