Define an equivalence relation on $\Bbb{Z}$ by $x\sim y$ if $x+2y$ is divisible by $3$

Question

How do I approach this problem? I know how to approach on the equivalence relation of triangles(finding $x\sim x$, finding if $x\sim y$ then $y \sim x$, and finding if $x \sim y$ and $y \sim z$, then $x \sim z$.

Answer 1

A hint: In ${\mathbb Z}$ the number $x+2y$ is divisible by $3$ iff $x-y$ is divisible by $3$.

Answer 2

In order to prove the relation is an equivalence relation, You need prove that it is:

• reflexive
  
  Is it always the case that $3 \mid (x + 2x)$.

• symmetry
  
  Is it always the case that if $3\mid (x + 2y)$, then $3\mid (y + 2x)$?

• transitivity
  
  Is it always the case that if $3\mid (x + 2y)$ and $3\mid (y + 2z)$, then $3\mid (x + 2z)$?

If YES for ALL three properties, then the relation is an equivalence relation.

If you find ANY ONE property fails to hold for all x, y, z, then it fails to be an equivalence relation.