Prove the following:

$($a$)$ If $n$ is odd, then $n^2$ is odd $($by direct proof$)$

$($b$)$ The converse of the above statement is also true $($by contrapositive$)$

$($c$)$ If $n$ is a perfect square, then $n+2$ is not a perfect square $($by direct proof$)$

$($d$)$ If the product of two numbers is odd $($say $x^{**}y),$ then both numbers $x$ and $y$ should be

odd $($use any method$).$

$($e$)n$ is an integer; if $3n+2$ is even, then $n$ is even $($by contradiction$)$

$($f$)$ The inverse of the above statement is true $($use any method$)$