Problem 3 (20 points). For each fixed $n \in \mathbb{Z}$, consider the equivalence relation ~ on $\mathbb{Z}$ given by $a \sim b$ if and only if $a - b$ is a multiple of $n$. Let $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}/\sim$ denote the quotient set and $\pi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ the quotient map.
(a) (10 points) Show that ~ is an equivalence relation and describe $\mathbb{Z}/n\mathbb{Z}$ when $n = 0$, $n = 1$, and $n = 2$.
(b) (10 points) Define operations + and · on $\mathbb{Z}/n\mathbb{Z}$ such that the quotient map $\pi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ satisfies $\pi(a + b) = \pi(a) + \pi(b)$ and $\pi(a \cdot b) = \pi(a) \cdot \pi(b)$ for all $a, b \in \mathbb{Z}$.
You need to verify that the operations you define are well-defined and indeed satisfy the properties above.
