Prove that a relation is Equivalent where: $7\mid x+6y$ where x and y both come from a set of numbers (1,2,3.....10)

Question

Like I've mentioned, I need to prove that it is equivalent, and get all of its classes. Proving that it is Reflexive is super easy and I did it myself. Proving symmetry and transitivity is what bothers me. I don't even want to talk about classes, which I definitely do not know how to define.

Thanks for your help in advance

Answer 1

hint For transitivity

Assume that $$7\mid(x+6y)$$ and $$7\mid(y+6z)$$

then

$$x+6y=7k$$ and $$y+6z=7k'$$

So, the sum gives $$x+6z=7(k+k'-y)=7k''$$

Other approach

$$x+6y\equiv 0 \mod 7$$ $$y+6z\equiv 0 \mod 7$$ $$-7y\equiv 0 \mod 7$$ the sum gives $$x+6z\equiv 0 \mod 7$$ Done.

Answer 2

Hint

$\frac{x+6y}{7}\in \mathbb{Z}$ if and only if $\pm\frac{x+6y}{7}+k\in \mathbb{Z}$ for all $k\in \mathbb{Z}$. The right hand side is easier to use to show symmetry or transitivity.