How would I show this proposition.
$ x\equiv_my\rightarrow\frac{x}{r}=\frac{y}{r}$
I will make $\frac{x}{r}$ capital X because it is easier to write. And $\frac{y}{r}$ capital Y. These are the equivalence classes.
I did this
Let w be any object. Let $w \in X$ thus $x\equiv_mw$. Since the mod is equivalent relationship it is symmetric.
Thus $y\equiv_mx$ and since $w\in Y$, $y\equiv m_w$.
In conclusion
X=Y.
For those interested in the thing here it is.
Let $w\in X$. This means that w is actually one the equivalent classes of mod for example congruence modulo 5 will have 0,1,2,3,4 as it cases.
$x\equiv 1 \mod 5$ is what x subtract by 1 is divisible by 5.
So then you do $x\equiv w\mod (m)$. By the symetry $y\equiv x \mod(m)$ then by the transitivity
$y=w \mod(m)$.
Thus $w\in Y$
Thus X is subset of Y.
Now to types the rest later time.