(i) State and prove the Inclusion-Exclusion Principle.

(ii) Let $n>1$ be an integer. Denote by $\mathbb{Z} / n \mathbb{Z}$ the integers modulo $n$. Let $X$ be the set of all functions $f: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}$ such that for every $j \in \mathbb{Z} / n \mathbb{Z}, f(j)-f(j-1) \not \equiv j$ $(\bmod n)$. Show that

$$|X|= \begin{cases}(n-1)^{n}+1-n & \text { if } n \text { is odd } \\ (n-1)^{n}-1 & \text { if } n \text { is even }\end{cases}$$