Let $\mathbb{Z}$ be the set of integers $$\lbrace \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \rbrace.$$ Define a binary relation on $\mathbb{Z}$ by declaring that $a \equiv b$ if and only if $a - b$ is divisible by $4$.

(a) Is $\equiv$ an equivalence relation? Prove that it is, or explain which parts of the definition of equivalence relation do not hold.


(b) Describe the set of all integers $b$ such that $b \equiv 0$.


(c) Describe the set of all integers $b$ such that $b \equiv 1$.