Statements can be proved by contrapositive, contradiction or by induction.
- <em>2.21 and 2.23 are proved by contrapositive</em>
- <em>2.22 is proved by induction</em>
<u />
<u />
<u>2.21: If </u>
<u> is even, then n is even (By contrapositive)</u>
The contrapositive of the above statement is that:
<em>If n is odd, then </em>
<em> is odd</em>
Represent the value of n as:
, where 
Take the cube of both sides

Expand

Group
![n^3 =[ 8k^3 + 6k^2 + 6k] + 1](https://tex.z-dn.net/?f=n%5E3%20%3D%5B%208k%5E3%20%2B%206k%5E2%20%2B%206k%5D%20%2B%201)
Factor out 2
![n^3 =2[4k^3 + 3k^2 + 3k] + 1](https://tex.z-dn.net/?f=n%5E3%20%3D2%5B4k%5E3%20%2B%203k%5E2%20%2B%203k%5D%20%2B%201)
Assume w is an integer; where:

So, we have:

The constant term (i.e. 1) means that
is odd.
Hence, the statement has been proved by contrapositive.
<em>i.e. If n is odd, then </em>
<em> is odd</em>
<u />
<u>2.22 </u>
<u> is even, if and only if n is even</u>
We have:
<u />
<u />
Assume that:
for 
So, we have:

Open bracket


Factorize

The factor of 2 means that
is even.
<em>Hence, </em>
<em> is even, if and only if n is even </em>
<em />
<u />
<u>2.22: </u>
<u> and </u>
<u>, then </u>
<u />
To do this, we prove by contrapositive.
The contrapositive of the above statement is:
If
and
, then 
We have:

Substitute the values of s and t in: 



Hence, by contrapositive:
If
and
, then 
Read more about proofs at:
brainly.com/question/19643658