Originally Posted by

**MadSoulz** Course: Foundations of Higher Math

A relation $\displaystyle R$ is defined on $\displaystyle \mathbb{Z}$ by *x R y* if $\displaystyle 3x-7y$ is even. Prove that $\displaystyle R$ is an equivalence relation. Determine the distinct equivalence classes.

Reflexivity:

Let $\displaystyle x \in \mathbb{Z}$.

$\displaystyle 3x-7x=-4x=2(-2x)$ which is even, so *x R x.*

Symmetry:

Let $\displaystyle x,y \in \mathbb{Z}$.

Assume that *x R y* ,i.e. $\displaystyle 3x-7y=2a$, $\displaystyle a\in \mathbb{Z}$.

$\displaystyle 3x-10x-7y+10y=2a-10x+10y$

$\displaystyle 3y-7x=2a-10x+10y$

which is even, so *y R x.*

Is this method fine?