Using the words necessary and sufficient rewrite the statement The integer `n` is odd if and only if `n^2` is odd Also check whether the statement is true.

Explanation

Let p: interger x is odd.
q : inerger `x^2` is odd.
To show that 'p if and only q' is true, we check the validaity of the following statements:
(i) if p and q
(ii) if q then p
(i) x is odd, then `x=2 m+1` where m is an interger.
`rArr x^2=(2m+1)^2`
`=4m^2+4m+1`
`=2(2m^2+2m)+1`
`rArrx^2` is odd.
`therefore` 'if p then q' is true.
(ii) 'if q then p' will be proved by contrapositive method.
Let x is not odd.
`rArrx` is even.
`rArrx+2n`,for some interger n
`rArrx^2=4n^2`
`rArrx^2` is even interger.
`rArrx^2` is not an odd interger.
`therefore` if q then p'is true.
Therefore , 'p if and only if q' is true.