let r be a relation defined on ℤ as follows: for all m, n ε ℤ, m r n iff 3 | (m2 – n2). a) prove that r is an equivalence relation.

Answers

Answer 1

To prove that r is an equivalence relation, we need to show that it satisfies the following three properties: Reflexivity, symmetry and transitivity.

a) Proving reflexivity: For all m ε ℤ, we need to show that m r m, i.e., 3 | (m2 – m2) = 0.

Since 0 is divisible by 3, reflexivity holds.

b) Proving symmetry: For all m, n ε ℤ, we need to show that if m r n, then n r m. Suppose m r n, i.e., 3 | (m2 – n2).

This means that there exists an integer k such that m2 – n2 = 3k. Rearranging this equation, we get n2 – m2 = –3k.

Since –3k is also an integer, we have 3 | (n2 – m2), which implies that n r m. Therefore, symmetry holds.

c) Proving transitivity: For all m, n, and p ε ℤ, we need to show that if m r n and n r p, then m r p.

Suppose m r n and n r p, i.e., 3 | (m2 – n2) and 3 | (n2 – p2). This means that there exist integers k and l such that m2 – n2 = 3k and n2 – p2 = 3l. Adding these two equations, we get m2 – p2 = 3k + 3l = 3(k + l). Since k + l is also an integer, we have 3 | (m2 – p2), which implies that m r p.

Therefore, transitivity holds.Since r satisfies all three properties of an equivalence relation, we can conclude that r is indeed an equivalence relation.

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Related Questions

The perimeter of the triangular park shown on the right is 15x-2 what is the missing length

Answers

Answer:

missing length is 6x - 1

Step-by-step explanation:

To find missing length, add two known sides and subtract from perimeter

(15x - 2) - (4x + 4 + 5x - 5)

(15x - 2) - (9x - 1) = 6x - 1

Which of the following best describes the equation below?
y = -5x + 12
A.
nonlinear
B.
linear
C.
both linear and nonlinear
o
D.
neither linear nor nonlinear

Answers

Answer:

B. linear equation

Step-by-step explanation:

How can you tell that whether the equation is linear or nonlinear?

Using an Equation

Simplify the equation as closely as possible to the form of y = mx + b. Check to see if your equation has exponents. If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

y = -5x + 12 has no exponents that's why it is a linear equation.

Hope this helps:)

Having trouble solving this:

Answers

The answer will be - 2/5

Answer:

1/250

Step-by-step explanation:

0.4% in fraction form is 0.004

or 4/1000

4/1000 reduced to it's simplest form is 1/250 (divide numerator and denominator by 4)


help find value of y please

Answers

Answer:

The third option

Step-by-step explanation:

60+y=180

y=120

Answer:

y = 120

Step-by-step explanation:

How far is the star away from
the earth in AU?

Answers

its about 27k AU away

Answer:

6623 AU

Step-by-step explanation:

I assume the angle measure 0.000302 is in radians.

Draw a segment from the Sun to the Star.

Use the upper small triangle.

The angle at the Star measures 0.000151 radians.

The opposite leg measures 1 AU.

We need the hypotenuse which is the distance from the Star to Earth.

sin A = opp/hyp

sin 0.000151 rad = 1/x

x sin 0.000151 rad = 1

x = 1/(sin 0.000151 rad)

x = 6623

Answer: 6623 AU

Help please! Thank youuuuuuuuu

Answers

Answer:

3 3/5 meters (B)

Step-by-step explanation:

Since this is perimeter, you must find the "outside".

The top side: 1/4 * 4 = 1

Left side: 1/5 * 4 = 4/5

Bottom: 1

Right side: 4/5

Add them up!

1 + 1 + 4/5 + 4/5

2 8/5

3 3/5

B!

(3a+2.5b)^2=9a^2+6.25b^2+*
Please help before Thursday

Answers

Step-by-step explanation:

We have (x + y)² = (x + y)(x + y) = x² + 2xy + y².

Let x be 3a and y be 2.5b.

Therefore (3a + 2.5b)²

= (3a)² + 2(3a)(2.5b) + (2.5b)²

= 9a² + 15ab + 6.25b².

Hence the '*' sign should be replaced with 15ab.

Answer:

Given that, ( 3a + 2.5b )² = 9a² + 6.25b² + *

We know that, ( x + y )² = x² + y² + 2xy

According to the above problem, we get

( 3a + 2.5b )² = (3a)² + (2.5b)² + (2×3a×2.5b)

( 3a + 2.5b )² = 9a² + 6.25b² + 15ab

Therefore,

The sign " * " should be 15ab.

15ab is the right answer.

HELP PELASE ASAPppppp

Answers

Answer:D

Step-by-step explanation:

Northern Bank has two types of account. Both accounts pay compound interest. Cash savings account Interest 2.5% per annum Shares account Interest 3.5% per annum Ali invests £2000 in the cash savings account. Ben invests £1600 in the shares account. (a) Work out who will get the most interest by the end of 3 years. You must show all your working.

Answers

Answer: Ben will have the most interest.

Step-by-step explanation:

Ali after 3 years.

To use compound interest, the formula is:

= Investment * ( 1 + rate) ^ no. of periods

= 2,000 * ( 1 + 2.5%)³

= £‭2,153.78

Interest = Amount after 3 years - investment

= 2,153.78 - 2,000

= £153.78

Ben after 3 years.

= 1,600 * (1 + 3.5%)³

= £1,773.95

Interest = 1,773.93 - 1,600

= £173.93

Simplify.
[tex]\frac{8\sqrt{6mn} + 6\sqrt{8mn} }{2\sqrt{2mn} }[/tex]

Answers

It’s the first answer
I used Math-way :)

Other equipment totaled 33000. You earn a commission of 20% on every sale you make. If you put 35% of your earnings on this particular sale into a savings account what is the amount invested

Answers

9514 1404 393

Answer:

  2310

Step-by-step explanation:

Assuming "this particular sale" totals 33000 in its entirety, the commission is ...

  20% × 33000 = 6600

The amount of that going to the savings account is ...

  35% × 6600 = 2310

The amount invested is 2310.

can someone pls help me
write 3lgx + 2-lgy as a single logarithm​

Answers

Answer:

log(x^3*y^(-2))

Step-by-step explanation:

two log rules you need to know for this question:

alogb = log(a^b)

logx + logy = logxy

you can start by getting rid of the term in front of the log, or a

also is it 2 minus or 2 times negative? im assuming it's 2 times negative because it is more likely

log (x^3) + log (y^-2) = log (x^3*y^-2)

plato de Leonardo tiene 0.2 kg de arroz y el de su hermana Leticia tiene 0.3 kg si quieren que los dos tengan lo mismo y los juntan para después repartirse los en partes iguales cuánto tendrá cada plato​

Answers

Answer:

0.5 kg

Step-by-step explanation:

Order the values of the differences from least to greatest. First, write the fractions as decimals. StartFraction 4 Over 5 EndFraction is and Negative 1 and one-half is . Next, plot the numbers on the number line. The greatest number is . Using the students' names, the correct order is .

Answers

Complete Question:

Ms. Perez's biology class grew sunflowers to learn about plants. During the first week, the average sunflower grew 3 inches. The table shows the difference from the average for three students.

Order the values of the differences from least to greatest.

Student: Raj. Difference (in.): 4/5

Student: Clara. Difference (in.): -1 1/2

Student: Jacob. Difference (in.): 0.9

Answer:

[tex]-1.5[/tex], [tex]0.8[/tex] and [tex]0.9[/tex]

The greatest is: 0.9

By student names: Clara, Raj and Jacob

Step-by-step explanation:

See comment for complete question

Given

Differences:

[tex]\frac{4}{5}[/tex], [tex]-1\frac{1}{2}[/tex], [tex]0.9[/tex]

Required

Order from least to greatest

We have: [tex]\frac{4}{5}[/tex], [tex]-1\frac{1}{2}[/tex], [tex]0.9[/tex]

First convert fractions to decimal

[tex]\frac{4}{5} = 0.8[/tex]

[tex]-1\frac{1}{2}= -1.5[/tex]

[tex]-\frac{3}{4} = -0.75[/tex]

So, the differences become:

[tex]0.8[/tex], [tex]-1.5[/tex], [tex]0.9[/tex]

Now arrange from least to greatest

[tex]-1.5[/tex], [tex]0.8[/tex] and [tex]0.9[/tex]

From the above representation,

The greatest is: 0.9

By student names, we have:

Clara, Raj and Jacob

This is done by simply replacing the numbers with the student name.

See attachment for number line

Answer:

4/5 is 0.8 , -1 1/2 is -1.5 , The greatest number is 1.3 , mia, raj, clara, jacob

Step-by-step explanation:

Determine a quadratic equation in standard form that has each pair of roots.
X=5±√2/3

Answers

Step-by-step explanation:

you have to add 5 and 2 which is 7 and decide by 3 or estimate

Add 5 and 2 which is 7 and then divide by 3

Two hundred fifteen people have been invited to a reception. The men get a gift of candy that cost $ 6.95 each, and the women get a gift of flowers that cost $ 8.95 each. If the total cost for gifts was $ 1724.25, how many men and how many women were invited?

Answers

The answer is $2000 your welcome

Please help due today in 10 minutes

Answers

Answer:

I can't help it's past due date. sorry

Step-by-step explanation:

Write an equation in slope intercept form. m=6, b=-8

Answers

Answer:

y=6x-8

Step-by-step explanation:

y=mx+b where m=slope and b=y-intercept.

I need the answer, ASAP.

Answers

Answer:

See below.

Step-by-step explanation:

It is similar because the order of operations is the same for fractions and for whole numbers.

It is different because when you add or subtract fractions, you need a common denominator. When you deal with whole numbers, there are no denominators.

express 4x-1= 3 as a linear equation in two variables​

Answers

Hello there!

~I hope you and your family are staying safe and healthy!

[tex]4x - 1 = 3[/tex]

We need to start by saying that y = 3

[tex]4x - 1 = y[/tex] or [tex]4x - y = 1[/tex]

Thus, the equation above is in the two variable form where y = 3

Thus, the required equation is [tex]4x - y = 1[/tex]

Let me know if you have additional questions!

~Garebear

Which expression is equivalent to 5x² + 4x - 2?
a- (5x-1)(x+2)
b- (5x+1)(x-2)
c- (5x+1)(x+2)
d- none of these

Answers

Answer:

D because non of them work lol

Step-by-step explanation:

Hope this Helped

What is the value of x?

Answers

Answer:

jhg hstu35kk4t6ek

3tnk63e6oktelrt

Step-by-step explanation:

4n5kdti3eo5

Paul will pay for his new car in 36 monthly payments. If his car loan is for $20,538 then how much will Paul pay each month?

Answers

Answer:

570.50

Step-by-step explanation:

I think it's the correct answer maybe it is maybe it's not

Paul has acquired a car loan of amount $20,538 and is paying off the loan amount in 36 months.  

This means that Paul will be paying $570.50 every month till the completion of 36 months.

The given parameters for the calculation of the monthly installment:

Loan amount= $20,538 Total months to repay the loan= 36 months

Calculation of the monthly amount of Paul's loan:

[tex]\begin{aligned}\text{Monthly installment of loan}= \frac{\text{Total Amount of Loan Acquires}}{ \text{Total Time taken to Repay off the Loan}} \end{aligned}.[/tex]

[tex]\begin{aligned}\text{Monthly Installment of Loan}= \frac{\$20,538}{36}\end{aligned}[/tex]

[tex]\begin{aligned}\text{Monthly Installment of Loan}=\$570.50\end{aligned}[/tex]

Therefore, the monthly installment of paul will be $57.50.

To know more about the calculation of installment, refer to the link below:

https://brainly.com/question/84351

For a school fundraiser, Troy sold 28 bags of popcorn and 40 candy bars and made $282. Jake sold 17 bags of popcorn and 20 candy bars and made $160.50.

What was the cost of a bag popcorn?

A. $10.35
B. $6.00
C. $6.27
D. $6.50

Answers

Answer:

Cost of a bag popcorn = $6.5

Step-by-step explanation:

Given:

Number of popcorn bags sold by Troy = 28

Number of candy bar sold by Troy = 40

Number of popcorn bags sold by Jake = 17

Number of candy bar sold by Jake = 20

Amount Troy earn = $282

Amount Jake earn = $160.5

Find:

Cost of a bag popcorn

Computation:

Assume;

Cost of a bag popcorn = a

Cost of a candy bar = b

28a + 40b = 282..........eq1

17a + 20b = 160.5...............eq2

eq2 x 2

34a + 40b = 321.........eq3

eq3 - eq1

6a = 39

a = 6.5

Cost of a bag popcorn = $6.5

What is the area of this?

Answers

Answer:

Step-by-step explanation:

[tex]A=\frac{hb}{2}\\ \\ A=\frac{14(20)}{2}\\ \\ A=140ft^2[/tex]

Answer:

150 ft

Step-by-step explanation:

You use Pythagoras theorem to work out the base of the triangle.

<3

On a map, 90 miles is represented by 3 inches. A measure of inch on the map would represent miles.

Answers

Complete Question

On a map, 90 miles is represented by 3 inches. A measure of 1/2 inch on the map would represent miles.

Answer:

15 miles

Step-by-step explanation:

From the above question,

3 inches = 90 miles

1/2 inch = x miles

Cross Multiply

3 inches × x miles = 90 miles × 1/2 inch

x miles = 90 miles × 1/2 inch/3 inches

x miles = 15 miles

Therefore, a measure of 1/2 inch on the map would represent 15 miles

3a + b = 225
b = a - 15

A.) 3a + b = a -15
B.) 3a + a = 225
C.) 3a + a - 15 = 225
D.) b = 3a + b

Answers

Answer:

c

Step-by-step explanation:

9 2/3 - 4 1/5. ====???

Answers

Answer:

Step-by-step explanation:

5.46666666667

Answer:

5 7/15

Step-by-step explanation:

9 2/3-4 1/5 = 9 10/15-4 3/15=5 7/15

The Sugar Sweet Company is going to transport its sugar to market. It will cost $6500 to rent trucks, and it will cost an additional $250 for each ton of sugar
transported
Let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported. Write an equation relating C to S. Then use this
equation to find the total cost to transport 14 tons of sugar.

Answers

Answer:

$10000 Total cost to transport 14 tons

Step-by-step explanation:

The total cost C(S) = $6500 + ($250/ton)S.  This is a linear function with initial value $6500 and slope $250/ton.

C(14) = $6500 + ($250/ton)(14) = $10000 = Total cost to transport 14 tons

Answer:

hello! the answer is below!

Step-by-step explanation:

Let's write the equation.

6500 + 250s = c

Now put in our value for s.

6500 + 250 · 19 = 4750

The answers are

6500 + 250s = c

And

4750

Hope this helps! :)

What is 2/3 divided by 6/12? REDUCE ANSWER TO SIMPLEST FORM.

Answers

Answer:

1 1/3

Step-by-step explanation:

To divide fractions, multiply by the reciprocal.

So we have:

2/3 x 12/6 = 24/18 = 4/3 = 1 1/3

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