Please help me with this math problem :)

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

A cup contains 3 blue paper clips,2 red paper clips,and 5 silver paper clips.what is the probability of not drawing a red paper

Elis [28]

Short Answer 0.8

Remark
The total number of possible draws is
3 Blue + 2 Red + 5 silver = 10
There are two ways of solving this, both of which will be used later on.

Method 1
If you don't draw a read paper clip, that means you will draw a silver or blue paper clip.
$\frac{Silver + blue}{10} = \frac{3 + 5}{10} = \frac{8}{10} = \text{0.8}$

Method Two
The other way is to draw a red one and take that probability away from 1.
P(red) = $\frac{2}{10} = \text{0.2}$
1 - 0.2 = 0.8 Same answer.

4 0

3 years ago

Help me plz show work

FinnZ [79.3K]

Answer:

c. 2,320miles

Step-by-step explanation:

29 divided by 1/4 = 116

116*20=2,320

7 0

3 years ago

Only 3 alligators can Set at the round table

user100 [1]

What is the problem? I don't understand

8 0

3 years ago

GIVING BRAINLIEST I ONLY NEED LIKE 2 OR 3 SENTENCES!

Softa [21]

Answer:

Trying to use a subject pronoun like he in a place of pool if the substituting sounds all right then use the subjective form who whom on the other hand is an object pronoun objective case therefore it should be used whenever it will do the job of direct object indirect object or object of preposition

Step-by-step explanation:

7 0

3 years ago