Q.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.

Given statement: "The integer n is odd if and only if $n^2$ is odd."

A biconditional "p if and only if q" means p is sufficient for q (p ⇒ q) and p is necessary for q (q ⇒ p). This is restated using "necessary and sufficient" as: "p is necessary and sufficient for q."

Here $p$: n is odd; $q$: $n^2$ is odd.

Rewritten: "n being an odd integer is necessary and sufficient for $n^2$ to be odd."

Checking truth:

• Sufficiency (n odd ⇒ n² odd): If n is odd, n = 2k+1, so $n^2 = 4k^2+4k+1 = 2(2k^2+2k)+1$, which is odd. ✓ …

For "p if and only if q", rewrite as "p is necessary and sufficient for q". To verify truth, separately prove both directions: p ⇒ q (sufficiency) and q ⇒ p ( …

• Reversing necessary/sufficient (saying q is sufficient for p when checking the wrong direction). …