Question

Show that the following statement is true
"The integer n is even if an only if n² is even"

Answer 1

The given statement can be rewritten as:
"The necessary and sufficient condition for integer n to be even is n² must be even".

Let p and q be the following statements.
p: The integer n is even.
q: n^​2 is even.
The given statement is "p if and only if q".
​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"
"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.
Thus, we have:

\[n^2 = 4 m^2\] 

Here, n² is even.
Therefore, "if p, then q" is true.
The statement "if q, then p" is given by
"If n is an integer and n² is even, then n is even".
To check he validity of the statement, we will use the contrapositive method. So, let n be an integer. Then,
n is odd.
Here,

\[n = 2k + 1\]   for some integer k.

\[\Rightarrow\]  \[n^2 = 4 k^2 + 2k + 1\]

Then, n^​2 is an odd integer.
n^​2 is not an even integer.
Thus "if q, then p" and "p if and only if q" are true.