Determine the equivalence classes on the relation a - b is in H where H = {4k : k is in Z}

Question

Let H = {4k : k ∈ Z}. A relation R is defined on Z by aRb if a − b ∈ H.
(a) Show that R is an equivalence relation.
(b) Determine the distinct equivalence classes.

---

A)
aRa if a - a is in H. Thus a - a = 0 and 0 = 4*m where m = 0. so R is reflexive.
Assuming a - b is in H, we want that b - a is in H. Thus if a - b = c = 4k, b - a = -c = 4(-k) where c is in H. Thus R is symmetric.
Let aRb and bRc, we want to show aRc. Thus if a - b = x and b - c = y, a - c = (x+b)-(b-y) = x + y; where x,y are in H. Thus x+y must also be in H and R is transitive.

B)
Now how would I determine the equivalence classes?

Answer 1

Let's use $\sim$ instead of $R$ since this is an equivalence relation. By definition, the equivalence class containing $a\in H$ is $$[a] = \{b \in \mathbb{Z} : a\sim b\}$$ Suppose that $b \in [a]$. Then $a\sim b$ which is equivalent to $a-b \in H$. This happens precisely when $a-b = 4k$ for some $k \in \mathbb{Z}$. That is $b \equiv a \mod{4}$. Thus, $[a] = \{b \in \mathbb{Z} : b \equiv a \mod{4}\}$.