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Mathematical Reasoning — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsMathematical Reasoning1 Mark
Question
Show that the following statement is true
"The integer $n$ is even if an only if $n^2$ is even"
✓

Answer

The given stat em en $t$ 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 stat em en t 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,
$n=2 m$, where $m$ is an integer
$\Rightarrow n ^2=(2 m)^2$
$\Rightarrow n ^2=4 m^2$
$\Rightarrow n ^2$ 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 statemens, we will use contrapositive method.
So, let $n$ be an odd integer. Then,
n is odd
$\Rightarrow n =2 k +1$ for some integer k :
$\Rightarrow n ^2=(2 k +1)^2$
$\Rightarrow n^2=4 k^2+4 k+1$
$\Rightarrow n ^2$ is not an even integer
Thus, $n$ is not even $\Rightarrow n^2$ is not even
$\therefore$ if $q$, then $p^{-}$is true.
Hence, "p if and only if $q^{-}$is true.

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