Let be an equivalence relation on a set S, and let a, b e S. Show that two equivalence classes under ~ are either equal or disjoint, i.e. either [a] = [b] or [a] n [b] = 0.

Answers

Answer 1

Given, an equivalence relation ~ on a set S. Let a and b be two elements in the set S. Assuming that [a] and [b] are two equivalence classes under the equivalence relation ~. Now we need to prove that either [a] = [b] or [a] ∩ [b] = ∅ (disjoint).

Proof:If [a] and [b] are not equal, then there must be some element c in the intersection of the equivalence classes [a] and [b]. i.e, c belongs to [a] and c belongs to [b].Thus, [a] ∩ [b] is not empty.

Let x be an element in [a], then x~a, and a~c (since c belongs to [a]) and hence x~c. So, x belongs to [c] which implies that [a] is a subset of [c].Now, let y be an element in [b], then y~b, and b~c (since c belongs to [b]) and hence y~c. So, y belongs to [c] which implies that [b] is a subset of [c].Thus, both [a] and [b] are subsets of [c].

Therefore, if [a] and [b] are not equal, then [a] and [b] are both subsets of [c] and hence the intersection of [a] and [b] is not empty. Thus, [a] and [b] are not disjoint. Hence, the proof by contradiction.

Conversely, if [a] and [b] are disjoint, then [a] ∩ [b] = ∅. And thus, [a] is not equal to [b].Therefore, two equivalence classes under the equivalence relation ~ are either equal or disjoint.

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Related Questions

find the change-of-coordinates matrix from the basisB = {1-7t^2, -6 + t+43t^2, 1+6t} to the standard basis. Then write t^2 as a linear combination of the polynomials in B.

Answers

To find the change-of-coordinates matrix from basis B to the standard basis, we need to express the standard basis vectors as linear combinations of the vectors in B. Then, to write t^2 as a linear combination of the polynomials in B, we can use the change-of-coordinates matrix to transform t^2 into the coordinates with respect to B.

To find the change-of-coordinates matrix from basis B to the standard basis, we express the standard basis vectors as linear combinations of the vectors in B. Let's denote the standard basis vectors as e1, e2, and e3. We can write:

e1 = 1(1 - 7t^2) + 0(-6 + t + 43t^2) + 0(1 + 6t)

e2 = 0(1 - 7t^2) + 1(-6 + t + 43t^2) + 0(1 + 6t)

e3 = 0(1 - 7t^2) + 0(-6 + t + 43t^2) + 1(1 + 6t)

The coefficients in these equations give us the entries of the change-of-coordinates matrix.

To write t^2 as a linear combination of the polynomials in B, we can use the change-of-coordinates matrix. Let [t^2]_B represent the coordinates of t^2 with respect to B. Then, [t^2]_B = C[t^2]_std, where C is the change-of-coordinates matrix. We can solve this equation to find [t^2]_B.

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Randomly select a painted rock from a bag containing 4 purple rocks, 3 green rocks, 3 orange rocks, and 2 blue rocks.

Answers

Answer:

i got a orange

Step-by-step explanation:

I would choose blue rocks bestie

Prove the following:

(2cos x) / (cos 2x + 1) = sec x

Answers

The equation (2cos x) / (cos 2x + 1) = sec x is proven to be true.

We start with the expression on the left-hand side and simplify it step by step to show that it is equivalent to sec x.

(2cos x) / (cos 2x + 1) = (2cos x) / ((2cos^2 x - 1) + 1)     [Using the double-angle formula cos 2x = 2cos^2 x - 1]

= (2cos x) / 2cos^2 x   [Simplifying the numerator and denominator]

= cos x / cos^2 x   [Cancelling out the factor of 2]

Now, we can simplify the expression further using the identity sec x = 1 / cos x:

cos x / cos^2 x = 1 / cos x = sec x.

Therefore, we have proved that (2cos x) / (cos 2x + 1) is equal to sec x.

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The graph of a system of equations is
shown below.

a. (2,0) and (0,3)
b. (-40,-50)
c. the system has no solution
d. the system has infinitely many solutions

Answers

Answer:

The system has no solution.

Step-by-step explanation:

In order for the system to have a solution, both graphs must have an intercept to each others. We see in the picture that both graphs are parallel and do not have any interceptions which we don't know the solution to the system.

That means if graphs are parallel and have no interceptions, there are no solutions. The system of equations are for finding the interceptions of both graphs. But of course! Parallel lines do not intercept.

If you have any questions, feel free to ask.

The J.R. Simplot Company produces frozen French fries that are then sold to customers such as McDonald's. The "prime" line of fries has an average length of 6.00 inches with a standard deviation of 0.50 inch. To make sure that Simplot continues to meet the quality standard for "prime" fries, they plan to select a random sample of n = 100 fries each day. The quality analysts will compute the mean length for the sample. They want to establish limits on either side of the 6.00 inch mean so that the chance of the sample mean falling within the limits is 0.99. What should these limits be?

Answers

Answer:

(5.871 `; 6.127)

Step-by-step explanation:

Given :

Mean = 6

Standard deviation. σ = 0.5

Samplr size, n = 100

Zcritical at 99% confidence = 2.58

The confidence interval :

Mean ± margin of error

Margin of Error = Zcritical * σ / sqrt(n)

Margin of Error = 2.58 * 0.5/sqrt(100) = 0.129

Confidence interval :

Lower boundary = 6.00 - 0.129 = 5.871

Upper boundary = 6.00 + 0.129 = 6.129

(5.871 `; 6.127)

The volume of a square pyramid is 48 in^3 with a height of 4 inches. Find the length of one of the side of the base.

Answers

Answer:

6 inches

Step-by-step explanation:

area of square = 48 x 3 / 4 = 36 in^2

side = 6 in

Answer:

6 in

Step-by-step explanation:

The volume formula for a square-base pyramid of side length x and height h is V = (1/3)x²h.  We want to solve this for the base side length x:

Multiplying both sides by (3/h) results in:

3V

----- = x²       and so the side length, x, is √(3V/h).

 h

Substituting 4 in for h and 48 in³ for V, we get:  x = √(144 in³/4 in) = 6 in

Check:  Does the volume formula given above result in 48 in³ when x = 6 in and h = 4 in?

     

V = (1/3)x²h  = (1/3)(6 in)²(4 in) = 48 in³        YES:  x =  6 in is correct.

What Statistic information we can get from a histogram?

A-Standard Deviation

B-Interquartile Range

C-Mean

Answers

The answer should be B

Find the area of the figure

Answers

Answer:

A=153.94

Step-by-step explanation:

Write and solve an equation to find the missing dimension of the figure.

Answers

Answer:

15.5769230769

Step-by-step explanation:

8×13 = 104

1620÷104 = 15.5769230769

(q11) Find the center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively.

Answers

The center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively is located at the point (1.6, 2.2).

To find the center of mass of the system of objects, we need to consider the masses and positions of each object. Let's go through the steps to calculate the center of mass:

Assign variables to the masses and coordinates of the objects:

Object 1: Mass = 2, Coordinates = (-1, 2)

Object 2: Mass = 3, Coordinates = (1, 1)

Object 3: Mass = 5, Coordinates = (3, 3)

Calculate the total mass of the system by summing the masses of the objects:

Total mass = 2 + 3 + 5 = 10

Calculate the weighted average of the x-coordinates and y-coordinates to find the center of mass:

Center of mass (x-coordinate) = (m1 * x1 + m2 * x2 + m3 * x3) / Total mass

Center of mass (y-coordinate) = (m1 * y1 + m2 * y2 + m3 * y3) / Total mass

Substituting the values:

Center of mass (x-coordinate) = (2 * -1 + 3 * 1 + 5 * 3) / 10 = ( -2 + 3 + 15) / 10 = 16 / 10 = 1.6

Center of mass (y-coordinate) = (2 * 2 + 3 * 1 + 5 * 3) / 10 = (4 + 3 + 15) / 10 = 22 / 10 = 2.2

Therefore, the center of mass of the system of objects is located at the point (1.6, 2.2).

The center of mass represents the average position of the system, taking  into account the masses and positions of the objects. It is a point where the system can be considered to be balanced. Calculating the center of mass is important in various areas of physics, such as studying the motion, stability, and collisions of objects.

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gh¯¯¯¯¯¯ has endpoints g(−3, 2) and h(3, −2). find the coordinates of the midpoint of gh¯¯¯¯¯¯ . a. (−3, 0) b. (0, 2) c. (0, 0) d. (0, −2)

Answers

The coordinates of the midpoint of the line segment GH with endpoints G(-3, 2) and H(3, -2) are (0, 0). The correct option is (C).

To determine the coordinates of the midpoint of the line segment GH with endpoints G(-3, 2) and H(3, -2), we can use the midpoint formula.

The midpoint formula states that the coordinates of the midpoint (M) are given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2)

For GH, plugging in the coordinates, we have:

Midpoint (M) = ((-3 + 3) / 2, (2 + -2) / 2)

Midpoint (M) = (0, 0)

Therefore, the coordinates of the midpoint of GH are (0, 0), which corresponds to option c. (0, 0).

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My Samsung charger broke so I went to Five Below to get a new one. They were on sale for $15,35 (not so five below). There
was a 6% tax too. How much did the new charger cost?
( I don’t need the explanation, only the right answer)

Answers

Answer:

The answer to your question is 16.27

Explanation:

You would start with the equation,

15.35 + 6% = C (Cost)

15.35 · 6% = 0.921

15.35 + 0.921 = 16.271

Round the one down, since it is less than four, and you get 16.27

Hope this helps.

waffletowne

HELPPPPP
Directions: Find the slope of the lines graphed below.
1.
2.
3.
4.
5.
6.
Directions: Find the slope between the given two points.
7.(-1,-11) and (-6, -7)
8. (-7.-13) and (1, -5)
9. (8.3) and (-5,3)
10. (15, 7) and (3,-2)
11. (-5, -1) and (-5, -10)
12. (-12, 16) and (-4,-2)
Directions: Use slope to determine if lines PQ and RS are parallel, perpendicular, or neither.
13.P(-9,-4), (-7, -1), R(-2,5), S(-6, -1)
m(PO)
m(RS)
Types of Lines
PLEASE HELLPPPP

Answers

Answer:

7. -4/5

8. 1

9. 0

10. 3/4

11. undefined/nonlinear

12. -9/4

13. parallel

Determine the area and circumference of a circle with radius 12 cm.

Answers

The area of the circle is 452.16 cm², and the circumference is 75.36 cm.

To determine the area and circumference of a circle with a radius of 12 cm, we can use the formulas:

Area = π * r²

Circumference = 2 * π * r

The radius (r) is 12 cm, we can substitute this value into the formulas to find the area and circumference.

Area = π * (12 cm)²

      = π * 144 cm²

      ≈ 3.14 * 144 cm²

      ≈ 452.16 cm²

The area of the circle is approximately 452.16 square centimeters.

Circumference = 2 * π * 12 cm

                   = 2 * 3.14 * 12 cm

                   ≈ 75.36 cm

The circumference of the circle is approximately 75.36 centimeters.

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Which statement explains the type of function that is represented by the equation
y=x2 +9?

Answers

The answer is 28392729

wayfktgfydfug9ugi0b7ffiobv57f9pogasdfyuiohfdfhjkl

Answers

Answer:

Area of the circle = 572.3 square ft.

Step-by-step explanation:

Area of a circle is given by the formula,

Area of a circle = πr²

Here 'r' = Radius of the circle

Diameter of circle given in the picture = 27 ft

Radius of the circle = [tex]\frac{27}{2}[/tex]

                                = 13.5 ft

Area of the circle = π(13.5)²

                             = 3.14(13.5)²

                             = 572.265

                             ≈ 572.3 square ft

SOMEONE PLS HELP SHEHSJHSHS

Answers

Answer:

1. Quadrant II

2. Quadrant IV

3. Y-axis

Step-by-step explanation:

What is the volume of this sphere? Use a ~ 3.14 and round your answer to the nearest hundredth. 20 mm​

Answers

Answer:

4186.67 cubic mm=V

Step-by-step explanation:

The volume of a sphere can be found using the equation, [tex]\frac{4}{3} \pi r^{3}[/tex]. It gives us the diameter but we need the radius. To find the radius just divde the diameter by 2, so the radius of the sphere is 10mm.

[tex]\frac{4}{3} (3.14) (10^{3})[/tex]=V

[tex]\frac{4}{3} (3.14) (1000)[/tex]=V

[tex]\frac{4}{3}[/tex](3140)=V

4186.67 cubic mm=V

Please help.
Is algebra.

Answers

Answer:

1 is C

2 is B

3 is C

Step-by-step explanation:

Answer:

give em brainliest because i have no idea

Step-by-step explanation:

Simplify

(2½)² x (0,5)²​

Answers

Answer:

1.5625

Step-by-step explanation:

2½ = 5/2

(5/2)² = 25/4

25/4 = 6.25

0.5² = 0.25

6.25 x 0.25 = 1.5625

A function, f(x),represents the height of a plant x months after being planted. Students measure and record the height on a
Select the appropriate domain for this situation.
OA.
the set of all real numbers
OB.
the set of all integers
O c.
the set of all positive integers
OD
the set of all positive real numbers

Answers

Answer:

I think the answer would be A or B, the best would be B.

Step-by-step explanation:

K so since it is monthly, it would not be all positive data. So you are left with A or B. Integers are numbers that are positive and negative so since the data can be a range of positive and negative numbers, that is why the data set of all integers (b) is the best choice.

Your welcome :D

Factor completely x3 − 2x2 − 8x + 16.
(x + 2)(x2 + 8)
(x − 2)(x2 + 8)
(x + 2)(x2 − 8)
(x − 2)(x2 − 8)

Answers

The expression x^3 - 2x^2 - 8x + 16 can be factored as (x - 2)(x^2 - 8).

To factor the given expression x^3 - 2x^2 - 8x + 16, we can look for common factors or factor it using grouping. In this case, we can observe that the expression can be factored by grouping.

First, we can factor out a common factor of (x - 2) from the first two terms:

x^3 - 2x^2 - 8x + 16 = (x - 2)(x^2 - 8x - 8)

Now, we can further factor the quadratic expression (x^2 - 8x - 8) by factoring out a common factor of 8:

(x^2 - 8x - 8) = (x - 2)(x - 8)

Therefore, the complete factorization of the expression x^3 - 2x^2 - 8x + 16 is:

(x - 2)(x - 2)(x - 8) which can also be written as (x - 2)(x^2 - 8).

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1. A package of sticky notes is in the shape of
a parallelogram. The dimensions of one
sticky note are shown. What is the area of
one sticky note?
A. 248.4 cm?
B. 124.2 cm2
C. 62.1 cm2
D. 22.7 cm2

Answers

Answer:

B. 124.2

Step-by-step explanation:

Area= b*h

Area= 13.5*9.2

Area = 124.2

Please Note* = Multiply

At 5 am, the temperature was -10°F. By noon the temperature was 7°F, What Integer
represents the change in temperature from 5 am, to noon?

Answers

Answer:

The integer that represents the change in temperature is 17.

Step-by-step explanation:

7 ⅓ ÷ 225 =
answer this question plz

Answers

Answer:

0.03259..

Step-by-step explanation:

what are the answers to a, b, c, d?

Answers

Answer:

Step-by-step explanation:

Find the volume of each composite figure.

Answers

Answer:

630 in^3

Step-by-step explanation:

6 x 6 x 14 = 504

(0.5 x 3 x 6) x 14 = 126

504 + 126 = 630

what are the values of a and b? a = 2y and b = 3 a = and b = –3y a = 2y and b = –3 a = and b = 3y

Answers

The values of a and b are dependent on the equation. For example, in the equation a = 2y and b = 3, then a = 6 and b = 3. However, in the equation a = 2y and b = -3, then a = -6 and b = -3.

In the given options, the values of a and b are stated as a = 2y and b = –3. This means that the value of a is equal to twice the value of y (a = 2y), and the value of b is equal to -3. The other options do not match these conditions. It is important to note that without further context or information about the variable y, we cannot determine a specific value for a or b. The values provided only establish the relationship between a, b, and y as described in the option a = 2y and b = –3.

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9. What measure of central tendency is usually the preferred number by researchers for describing a group of scores?

10. Using the following data, calculate the standard deviation: 115, 125, 145, 150, 115, 125

Answers

The measure of central tendency that is usually the preferred number by researchers for describing a group of scores is the mean.

The measure of central tendency that is usually the preferred number by researchers for describing a group of scores is the mean. The mean provides the average score for a given set of data. It is calculated by summing up all the scores and dividing by the total number of scores. While other measures of central tendency, such as the median and mode, are also useful, the mean provides the most comprehensive picture of the data.


10. To calculate the standard deviation for the following data: 115, 125, 145, 150, 115, 125, you can use the following formula:

σ = √[(Σ(x - μ)²) / N]

Where:
σ = standard deviation
Σ = sum of
x = each score
μ = mean of all the scores
N = total number of scores

To find the standard deviation, first find the mean of the scores:

115 + 125 + 145 + 150 + 115 + 125 = 775
Total number of scores = 6
Mean = 775 / 6 = 129.17

Now, subtract the mean from each score, square the result, and add up all the squared differences:

(115 - 129.17)² + (125 - 129.17)² + (145 - 129.17)² + (150 - 129.17)² + (115 - 129.17)² + (125 - 129.17)² = 4666.79

Then, divide the sum of squared differences by the total number of scores and take the square root:

σ = √(4666.79 / 6) = 17.20


Mean = (115 + 125 + 145 + 150 + 115 + 125) / 6 = 129.17

Standard Deviation = √[ {(115 - 129.17)² + (125 - 129.17)² + (145 - 129.17)² + (150 - 129.17)² + (115 - 129.17)² + (125 - 129.17)² } / 6 ]

= √[4666.79 / 6]

= 17.20

Therefore, the standard deviation of the given data is 17.20.

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Simplify the following expressions to have fewer terms (MIDDLE SCHOOL)

5x-3+(4x-6)+2

4y+3-(8x-2)

Answers

Answer:

Install math calculator

Thank you

Answer:

x=1\9

Step-by-step explanation:

first part:

5x-3+4x+2

5x+4x-3+2

9x-1  

9x=1

x=1\9 answer.

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Why was it necessary to calculate an error ellipsoid to accurately predict the satellite's return to the planet's surface?Which statement best summarizes this excerpt? Katherine Goble sat with the engineers to discuss the requirements for the lectures and reports. She listened closely and asked questions to deepen her understanding of the way things work so she could create a more accurate model. Katherine Goble listened to the engineers and asked thoughtful questions, wanting to gain a more extensive knowledge of the subject and complete her task with precision.Katherine Goble met with the engineers to askquestions about what would be expected of her in her new job and to gain information that would help her advance her career.Katherine Goble and the engineers met to discuss the specifics of the task she was asked to complete. She asked questions that showed her immense talent and A tank on the road roller is filled with water. 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