Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

\(\begin{aligned}
& f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\
(4-n) / 2 & \text { if } n \text { is even }\end{cases} \\
& f(n)=\{(1,1),(2,1),(3,2),(4,0),(5,3),(6,-1), \ldots\}
\end{aligned}\)
\(\therefore \mathrm{f}(\mathrm{n})\) is many one and onto function
\(\begin{aligned}
& \mathrm{g}(\mathrm{n})=\left\{\begin{array}{cc}
3+2 \mathrm{n} & \text { if } \mathrm{n} \geq 0 \\
-2 \mathrm{n} & \text { if } \mathrm{n} <0
\end{array}\right. \\
& \mathrm{g}(\mathrm{n})=\{(-3,6),(-2,4),(-1,2),(0,3),(1,5),(2,7),(3,9),(4,15), \ldots\}
\end{aligned}\)
\(\therefore \mathrm{g}(\mathrm{n})\) is one-one and into function
\(\mathrm{f}(\mathrm{~g}(\mathrm{n}))=2+\mathrm{n}, \mathrm{n} \in \mathrm{~N}\)
fog is one-one and into
\(g(f(n))= \begin{cases}4+n & \text { if } n \text { is odd natural number } \\ 7-n & \text { if } n=2,4 \\ n-4 & \text { if } n \text { is even natural number and } n \geq 6\end{cases}\)
\(g(f(2))=g(f(1))=5\)
\(\therefore\) gof is many one and into