How to get started on this proving this set?

Question

Let $R$ denote the set of all round integers and let $S$ denote the set of strange integers. Show that $R \cup S = \mathbb{Z}$.

An integer $n$ is called strange iff there exist an integer $k$ such that $n=3k+1$.
An integer $n$ is called round iff $3\mid n$.
An integer $n$ is called weird iff $n+1$ is round.

I know that this statement is false. I'm having trouble trying to create a salvage for this. I know that I need all 3 of these definitions in order for me to prove it is true for all integers. Can you help me get started?

It is weird. But we now have proved that $R\cup S\ne \mathbb Z$. — , Nov 06 '16 at 05:08
I have to prove that the statement is true with minor changes. Like I could add the definition of even (n=2k) or odd (n=2k+1) — ash, Nov 06 '16 at 05:28
Well its not true because it doesn't include any of the 3k-1. Either redefine strange as $3k\pm 1$ or define $3k-1$ terms and add a third set. — fleablood, Nov 06 '16 at 05:48

score 0 · Answer 1 · answered Nov 06 '16 at 05:35

Hint: Note that:

$n$ is strange $\iff 3|(n-1)$

$n$ is round $\iff 3|n$

$n$ is weird $\iff 3|(n+1)$

Then, $R=\{$strange numbers$\}$, $S=\{$round numbers$\}$ and $T=\{$weird numbers$\}$ are a partition of $\Bbb{Z}$, that is, $\Bbb{Z} = R \cup S \cup T$ and $R\cap S=R\cap T=S\cap T= \emptyset$.

How to get started on this proving this set?

1 Answers1