Question

Show that the following statement is true

"The integer n is even if and only if $n^2$ is even"

Answer 1

The given statement can be re-written as

"The necessary and sufficient condition that the integer n is even is $n^2$ must be even

Let p and q be the statements given by

p : the integer n is even.

q : $n^2$ is even.

The given statement is

" p if and only if q"

In order to check its validity, we have to check the validity of the following statements

(i) "If p, then q"

(ii) "If q, then p"

Checking the validity of "if p, then q" :

The statement " if p, then q " is given by :

"If the integer n is even, then $n^2$ is even "

Let us assume that n is even. Then ,

n = 2m, where m is an integer

$\Rightarrow n^2=(2m{)}^2$

$\Rightarrow n^2=4m^2$

$\Rightarrow n^2$ is an even integer

Thus, n is even $\Rightarrow n^2$ is even

$\therefore$ "if p, then q" is true.

Checking the validity of "if q , then p" :

" If n is an integer and $n^2$ is even, then n is even"

To check the validity of this statements, we will use contrapositive method.

So, let n be an odd integer. Then ,

n is odd

$\Rightarrow$ n = 2k +1 for some integer k

$\Rightarrow$ n^2 = (2k+1)^2\)

$\Rightarrow n^2=4k^2+4k+1$

$\Rightarrow n^2$ is not an even integer.

Thus, n is not even $\Rightarrow n^2$ is not even

$\therefore$ "if p , then q' is true.

Hence, "p if and only if q" is true.