Question

Let R be a relation on ℤ given by xRy if and only if x²-y² is divisible by 3. Show that this relation is an equivalence relation and find its corresponding equivalence classes.

Answer 1

First we show that given relation R is an equivalence relation:

$1.\ \forall x\ xRx: \quad 3 \mid 0 \implies 3 \mid x^2 - x^2$

$2.\ xRy \implies yRx : \quad 3 \mid x^2 - y^2 \implies 3 \mid-(x^2 - y^2) \implies 3 \mid y^2 - x^2$

$3.\ xRy, yRz \implies xRz: \ 3 \mid x^2 - y^2 ,\ 3 \mid y^2 - z^2 \implies \\ \implies 3 \mid (x^2 - y^2) + (y^2 - z^2) \implies 3 \mid x^2 - z^2$

Hence by definition it is indeed an equivalence relation.

Now we find the equivalence classes:

$3 \mid x^2 - y^2 \\ 3 \mid (x-y)(x+y) \\ 3 \mid x - y \quad or \quad 3 \mid x+y$

$x = y + 3k \quad or \quad x = -y + 3k, \quad k \in \mathbb{Z}$

We can see from here, that for every $y \in \mathbb{Z}$ the corresponding equivalence class is:

$[y] = \{ x \in \mathbb{Z} \mid x = \pm y + 3k, \ k \in \mathbb{Z} \}$