For each statement, write what would be assumed and what would be proven in a proof by contrapositive of the statement. Then write what would be assumed and what would be proven in a proof by contradiction of the statement.

a. If x and y are a pair of consecutive integers, then x and y have opposite parity.
b. For all integers n, if n² is odd, then n is also odd.

Answers

Answer 1

Answer:

a)

Given Statement - If x and y are a pair of consecutive integers, then x and y have opposite parity.

Proof by Contrapositive:

Assumed statement: Suppose that integers x and y do not have opposite parity.

Proven Statement: x and y are not a pair of consecutive integers.

Proof -

x = 2u₁ , y = 2u₂

Then

(x, x+1) = (2u₁ , 2u₁ + 1) = (Even, odd)

If y = 2u₁ + 1

Not possible

⇒x and y are not a pair of consecutive integers.

Hence proved.

Proof by Contradiction:

Assumed statement: Suppose x and y are not a pair of consecutive integers.

Proven Statement: Suppose x and y do not have opposite parity.

Proof -

If x and y are not a pair of consecutive integers.

⇒ either x and y are odd or even

If x and y are odd

⇒x and y have same parity

Contradiction

If x and y are even

⇒x and y have same parity

Contradiction

(b)

Proof by Contrapositive:

Assumed statement: Let n be an integer such that n is not odd (i.e. n is an even integer)

Proven Statement: n² is not odd (i.e n² is even)

Proof -

Let n is even

⇒n = 2m

⇒n² = (2m)² = 4m²

⇒n² is even

Hence proved.

Proof by Contradiction:

Assumed statement: Let n be an integer such that n² be odd.

Proven Statement:  suppose that n is not odd (i.e n is even)

Proof -

Let n² is odd

⇒n² is even

⇒n² = 2m

⇒2 | n²

⇒2 | n

⇒n = 2x

⇒ n is even

Contradiction


Related Questions

Find the equation of the line passing through the points (-2,8) and (8,- 7). Write the equation in
slope-intercept form.

Answers

Answer:

Step-by-step explanation:

[tex]y=mx+b\\ \\ m=\frac{-7-8}{8-\ -2}=\frac{-15}{10}=\ -\frac{3}{2}\\ \\ y=\ -\frac{3x}{2}+b,\ (-2,8)\\ \\ 8=\ -\frac{3(-2)}{2}+b\\ \\ 8=3+b,\ b=5\\ \\ y=\frac{10-3x}{2}[/tex]

Answer:   y  = - 3/2x + 5

Step-by-step explanation:

To write the equation in slope intercept form, you will have to find the slope and the y-intercept.

(-2,8) (8,-7)  

To find the slope using these points, you will find the difference in the y coordinates and divide it by the difference in the x coordinates.

-7 - 8 = - 15

8 - (-2) = 10

   Slope:   -15/10   =  -3/2  

The slope is  -3/2

Now using the formula y=mx+b  , use one of the given points and the slope to solve for b , which is the y-intercept.

     8  = -3/2(-2) + b

    8 = 3 + b

   -3     -3

    b = 5

What is 9% of 2 trillion

Answers

Answer: 180000000000

Step-by-step explanation:

Answer:

180000000000

Step-by-step explanation:

a circle with the diameter of 40 inches. what is the area?

Answers

Answer:

A= 1256

Step-by-step explanation:

The formula for area of a circle is [tex]A=\pi r^{2}[/tex]

1. Diameter is DOUBLE of radius, so the diameter must be divided by 2. 40/2 = 20.

2. Then, plug in the values into the area formula. A= 3.14[tex](20^{2})[/tex]

3. Solve. A= 1256. (Remember that 20 squared is 20x20)

I WILL GIVE BRAINLIEST!!!! ~PLEASE HELP (all the answers for all the questions please for the second and first picture)

Answers

Answer:

Question 6: I know that (8 + 9) + 3 is equal to 8 + (9 + 3) because of the associative property states that you can change the grouping of addends in any order you want. The benefit to using this property is that you can first add numbers that you can combine more easily. For example, (743 + 51) + 49 can be written as 743 + (51 + 49) or 743 + 100. It is more straightforward to add the numbers that equal 100 first and then add the final number.

Question 3: -3(4x + 3) + 5x [Given]; -12x - 9 + 5x [Distributive Property]; -12x + 5x - 9 [Commutative Property]; -7x - 9 [Combine like-terms]. [tex]\frac{1}{2}[/tex] (6k + 10) + 5k [Given]; 3k + 5 + 5k [Distributive Property]; 3k + 5k + 5 [Commutative Property]; 8k + 5 [ Combine like-terms].

Question 4: 4(2 - 5) - 8 = 4 * (-3) - 8 = -12 - 8 = -20; 4 * 2 - 28 = 8 - 28 = -20.

Question 5: 15 * 3 - 2(2 * 3 + 6) = 45 - 2(6 + 6) = 45 - 2(12) = 45 - 24 = 21; 11 * 3 - 12 = 33 - 12 = 21.

Please help me one more time

Answers

Answer: 24

Yes (x-2) is a factor. There is a remainder.

Step-by-step explanation:

2³-8•2²+14•2-4

8-8•4+14•2-4

0•4+14•2-4

0+14•2-4

14•2-4

28-4

24

Hope this helps if so plz mark me brainly

300x + 20 + 5x + 10 -40x2 =​

Answers

Step-by-step explanation:

[tex]305x + 30 - 80 \\ 305x - 50 \\ 305x = 50 \\ x = \frac{305}{50} \\ x = \frac{61}{10} [/tex]

In a recent survey of 1,000 randomly selected registered voters in the town of Armstrong, 803
are in favor of installing bicycle safety signs. Based on the survey results, approximately how
many of Armstrong's 16,092 registered voters are in favor of installing bicycle safety signs?
A. 803
B. 3,170
C. 12,922
D. 15,092

Answers

Answer:

12,922

Step-by-step explanation:

estimate = 16,092 * 803/1,000 = 12,922

help plz!! Marking most Brainly :)))

Answers

Answer: A

Step-by-step explanation:

12 black cards / 20 cards in all.

20 x 5 = 100, 12 x 5 = 60

brian is buying cans of baked beans for a camping trip.each can is 15 ounces.if he buys 7 cans,about how many pounds of beans is he buying?
2 pounds
5 pounds
7 pounds
10 pounds​

Answers

Answer:

about 7 pounds

Step-by-step explanation:

NOTE: 16 ounces equals a pound

so, we would do 15x7 which equals 105 ounces. then we would do 105/16 since 105 is the ounces in total and 16 ounces=1 pound. so our answer would equal to 6.5625. which rounded off would be about 7 pounds. I hope this is correct /helps!:)

Four crayons are packaged in a row, three primary colors are touching. If neither is red or blue, which crayon is first

Answers

Answer:

yellow

Step-by-step explanation:

SHAYNA FREEL PLANS TO BUY $17.45 WORTH OF STATIONERY SUPPLIES. IF THE SALES TAX IS 13%, HOW MUCH WILL SHAYNA PAY?

Answers

9514 1404 393

Answer:

  $19.72

Step-by-step explanation:

The sales tax is 13% of $17.45:

  0.13 × $17.45 = $2.27 . . . . . . rounded (up) to the nearest cent

Then the total Shayna will pay is ...

  $17.45 +2.27 = $19.72

-a^2+7b^4-2c^3 if a= -4 b= -2 and c=-3

Answers

Step-by-step explanation:

-(-4)²+7(-2)⁴-2(-3)³

=16+7(16)-2(-27)

=15+112+54

=181

The evaluation of the expression is 150.

What is an expression?

In mathematics, expression is the combination of variables with the application of operations and prescribed rules. It might take the shape of an equation, some numbers, etc.

The given expression is:

-a² + 7b⁴  -2c³

Substituting, a= -4 b= -2 and c=-3

-a² + 7b⁴  -2c³

= -(-4)² + 7(-2)⁴  -2(-3)³

= -16 + 7 x 16 + 54

= 150

Therefore, the value of the expression is 150.

To learn more about the expression;

brainly.com/question/24242989

#SPJ2

A cab company charges a $3.95 flat fee for rides up to 15 miles. For rides over 15 miles, the company charges the $3.95 flat fee and $1.25 per mile for every mile over 15 miles up to 75 miles. For rides over 75 miles the company charges a flat fee of $78.95 and $0.75 per mile over 75 miles. Using this information, write an equation that represents the total cost of the trip as a function of the number of miles traveled. (Remember: Identify your input and output variables, and use proper notation – I recommend you handwrite your function and add a picture in to this document)

Answers

Answer:

There are three intervals depending on  the mileage:

0 - 15 miles15 - 75 milesMore than 75 miles

For each interval we have different cost and equation

Let miles = x, cost  = y, then the equations are:

0 < x ≤ 15       ⇒ y = 3.9515 < x ≤ 75     ⇒ y = 1.25x + 3.95x > 75             ⇒ y = 0.85x + 78.95

Step-by-step explanation:0 - 15 miles

15 - 75 miles

More than 75 miles

For each interval we have different cost and equation

Let miles = x, cost  = y, then the equations are:

0 < x ≤ 15       ⇒ y = 3.95

15 < x ≤ 75     ⇒ y = 1.25x + 3.95

x > 75             ⇒ y = 0.85x + 78.95

13x+7=215
×=
Solve the equation and enter the value of x

Answers

Answer:

x=16, have a great day

Step-by-step explanation:

Answer:

x = 16

Step-by-step explanation:

13x + 7 = 215

13x = 208 (Subtract 7 from both sides.)

x = 16 (Divide both sides by 13)

Please help right now which of the following is a major challenge to volcano preparedness​

Answers

Answer:

D or C I think

Step-by-step explanation:

How do someone overrun so much lava and disaster kits are eated by lava so

The answer is (C)
Hope this helps!

Mike bought 46 dozen eggs from the grocery store to bake some cakes how many eggs did mike buy

Answers

Answer:

552 eggs

Step-by-step explanation:

A dozen is equal to 12

Solve:

Multiply 46 by 12

Find the missing length. If needed, round to the nearest tenth.
B
6
7
5
Х
A
The value of x is

Answers

Answer:

x=5.8

Step-by-step explanation:

5/6=x/7

cross multiply

6x=35

x=35/6

x=5.8

Solve the Antiderivative.​

Answers

Ln ( | Sec (e^x + π) + tan ( e^x +π ) | )

Answer:

I got the same as the top so I think that's the answer

Which of the following expressions are equivalent to -9/j?
A: 9/-j
B: - -9/j
C: none of the above

Answers

Answer:

A

Step-by-step explanation:

-9/j  is the same as 9/-j

Write the fraction in simplest form. *
8/12

Answers

Answer:

2/3

Explanation:

/////////////

em um campeonato de ciclismo um atleta precisou percorrer 54 000 m para a faixa de Chegada Sabendo que Ele demorou 7 2000 Segundos a Concluir a Corrida qual foi dua Velocidade Media em Km/h?​

Answers

Answer:

Average speed in km / h = 2.7 km/h

Step-by-step explanation:

Given:

Total distance = 54,000 m

Total time = 72,000 sec

Find:

Average speed in km / h

Computation:

Average speed = Distance / Time

Average speed = 54,000 / 72,000

Average speed = 0.75 m/s

1 m/s = 3.6 km/h

So,

Average speed in km / h = 0.75 x 3.6

Average speed in km / h = 2.7 km/h

To complete her holiday wrapping, Bella needs identical
strips of ribbon measuring 3.5 feet each. She has 85 feet of
ribbon in total. About how many pieces of ribbon can she
cut from this long strip?

Answers

Answer:

24

Step-by-step explanation:

Answer:

The answer would be At least 20 box.

Step-by-step explanation:

i anwerd it in the lesson and got it right. Hope this helps. :)

Tell whether the ordered pair is a solution to the system of linear equations. 1 point
(-2,-1); 2x – 4y= -8
6x + 5y = -7
Yes
No
Maybe
Tell whether the ordered nair is a solution to the

Answers

Answer:

Tell whether the ordered pair is a solution to the system of linear equations. 1 point

(-2,-1); 2x – 4y= -8

6x + 5y = -7

No

All the fractions that are equivalent to 4/5 A. 6/10 B. 8/10 C. 8/15 D. 10/15 E. 12/15

Answers

Answer:

B.8/10,E.12/15

Step-by-step explanation:

4/5×2=8/10

4/5×3=12/15


Can anyone help with condensing this?

Answers

Answer:

log 16x^7.

Step-by-step explanation:

log 4 + 2 log 2 + 7 log x

= log4 + log 2^2 + log x^7

= log (4*4*x^7)

= log 16x^7.

1. Clare has $150 in her bank account. She buys a bike for $200. What is Clare’s account balance now?

2. If Clare earns $75 the next week from delivering newspapers and deposits it in her account, what will her account balance be then?

Answers

For answer 1. -50
For answer 2. 25

Answer:

1. -$50 2. $25

Step-by-step explanation:

150-200= -50

-50+75=25

HELP PLEASE Find the surface area of the 3D shape.

Answers

Answer:

88 cm^2

Step-by-step explanation:

Answer:

Its totally (A) 88cm2

Step-by-step explanation:

Cause i got it right

Please help me I can’t figure out how to solve

Answers

i don’t know how to do it,, but use photomath, it’s perfect for these type of questions

Answer: The answer is 20

Step-by-step explanation:

x-9.5=10.5

  +9.5  +9.5

10.5+9.5=20

20-9.5=10.5

I hope this helps if so please mark brainliest

What would the figure below look like after a reflection across the x-axis?

Answers

third one is across the x-axis

A used automobile dealership recently reduced the price of a used compact car by 10​%. If the price of the car before discount was ​$18,100​, find the discount and the new price.
The discount in price is ​$
​Thus, the new price is ​$


Answers

Answer:

A. $1,810, B. $16,290

Step-by-step explanation:

A. $18,100 * 0.10 = $1,810

B. $18,100 - $1810= $16,290

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