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Problems 2.21, 2.22, 2.23

RoseWind [281]

Statements can be proved by contrapositive, contradiction or by induction.

2.21 and 2.23 are proved by contrapositive
2.22 is proved by induction

2.21: If $n^3$ is even, then n is even (By contrapositive)

The contrapositive of the above statement is that:

If n is odd, then $n^3$ is odd

Represent the value of n as:

$n = 2k + 1$ , where $k \ge 0$

Take the cube of both sides

$n^3 = (2k + 1)^3$

Expand

$n^3 = 8k^3 + 6k^2 + 6k + 1$

Group

$n^3 =[ 8k^3 + 6k^2 + 6k] + 1$

Factor out 2

$n^3 =2[4k^3 + 3k^2 + 3k] + 1$

Assume w is an integer; where:

$w =4k^3 + 3k^2 + 3k$

So, we have:

$n^3 =2w + 1$

The constant term (i.e. 1) means that $n^3$ is odd.

Hence, the statement has been proved by contrapositive.

i.e. If n is odd, then $n^3$ is odd

2.22 $3n + 4$ is even, if and only if n is even

We have: $3n + 4$

Assume that: $n = 2k + 2$ for $k \ge 0$

So, we have:

$3n + 4 = 3(2k + 2) + 4$

Open bracket

$3n + 4 = 6k + 6 + 4$

$3n + 4 = 6k + 10$

Factorize

$3n + 4 = 2(3k + 5)$

The factor of 2 means that $3n + 4$ is even.

Hence, $3n + 4$ is even, if and only if n is even

2.22: $s \ne -1$ and $t \ne -1$ , then $s + t + st \ne -1$

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

We have:

$s + t + st = -1$

Substitute the values of s and t in: $s + t + st = -1$

$-1 -1 -1 \times -1 = -1$

$-1 -1 + 1 = -1$

$-1 = -1$

Hence, by contrapositive:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

Read more about proofs at:

brainly.com/question/19643658

7 0

3 years ago

As air rises, it cools and provides abundant rain in the tropics. True or False

Korolek [52]

Answer:

true

when air rises it falls rain

8 0

3 years ago

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What is 3/4 divided by 5 1/5?

vampirchik [111]

15/104 would be your answer.

7 0

3 years ago

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Jane has a checkbook balance of $68.00. She then writes two checks, one for $5.00 and one for $62.50. She also deposits $75.00.

andrew-mc [135]

The answer here would be C. ON/C 6 8 – 5 – 6 2 . 5 0 + 7 5 =

Following her keypresses, she has to turn the calculator on by pressing "ON/C". Entering her current balance, she presses 6 then 8, subtracting her 2 checks by entering "-" then 5, and again for 62.50. Lastly, she adds the amount deposited which is 75. Pressing the "=" sign then tells her how much her new balance is.

8 0

3 years ago

What is the solution to the above system of equations?

EleoNora [17]

Answer: A. (3,-2)

Step-by-step explanation:

Solve by Substitution :

// Solve equation [2] for the variable x

[2] 4x = 5y + 22

[2] x = 5y/4 + 11/2

// Plug this in for variable x in equation [1]

[1] 2•(5y/4+11/2) + 5y = -4

[1] 15y/2 = -15

[1] 15y = -30

// Solve equation [1] for the variable y

[1] 15y = - 30

[1] y = - 2

// By now we know this much :

x = 5y/4+11/2

y = -2

// Use the y value to solve for x

x = (5/4)(-2)+11/2 = 3

Solution :

{x,y} = {3,-2}

7 0

4 years ago

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