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Show that the Following Statement is True "The Integer N is Even If an Only If N2 is Even" - Mathematics

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

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

Let and q be the following statements.
p: The integer n is even.
qn​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 is even.

\[n = 2m\] ,where is an integer.
Thus, we have:
\[n^2 = 4 m^2\] 

Here, nis 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,
is odd.

\[n = 2k + 1\]   for some integer k.
\[\Rightarrow\]  \[n^2 = 4 k^2 + 2k + 1\]

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

Concept: Mathematical Reasoning - Difference Between Contradiction, Converse and Contrapositive
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RD Sharma Class 11 Mathematics Textbook
Chapter 31 Mathematical reasoning
Exercise 31.6 | Q 5 | Page 29
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