Equivalence Relations Proof dealing 3 dividing x + y

Question

Consider the relation $S$ on the Natural Numbers defined by $\quad x\,S\,y\quad$ if $3$ divides $\quad x + y.\quad$ Prove $S$ is not an equivalence relation.

I know an equivalence relation is one that is reflexive, symmetric, and transitive. I believe that S does not satisfy the reflexive property. For example, the element (2,2) would not be an element of S.

Is this correct? Also I am unsure how to state that formerly.

Thanks for the Help.

Answer 1

I don't have enough points to comment, or I would make this into a comment:

You need to show:

i) Every element in the set is related to itself; in this case, $x\sim x$, meaning

$x+x$ is divisible by $3$.

ii) If $x\sim y$, then $y\sim x$, i.e., if $x+y$ is divisible by $3$, then so is $y+x$.

iii) If $x\sim y$ and $y\sim z$, then $x\sim z$.

Can you take it from here?

Answer 2

$3$ not necessarily divides $2x$.