# Math Help - equivalence classes

1. ## equivalence classes

I'm having trouble proving the distributive law for [r] [s] [t] in Zn (the integers mod n)

i.e. [r]([s] + [t]) = [r][s] + [r][t]

I did this so far.
(r+m1*n)[(s+m2*n)+(t+m3*n)] = (r+m1*n)(s+m2*n)+(r+m3*n)(t+m4*n)

2. Hello,

(r+m1*n)[(s+m2*n)+(t+m3*n)]= (r+m1*n)(s+m2*n)+(r+m1*n)(t+m3*n)
*modification in red*

Now, huum... you can see, if you develop again, you have [r][s]+[r][t]

3. [r] = r+mn, the m could be different for each [r], so the first [r] could have a different m than the second.

4. Here is an example how you prove it is well-defined.
Say you wanted to prove $[a]+[b]=[a+b]$.

Say $[a]=[c]$ and $[b]=[d]$.
This means $a\equiv c ~ (n)$ and $b\equiv d ~ (n)$.
Thus, $a+b\equiv c+d ~ (n)$ so $[a+c]=[b+d]$.
This shows that $+$ is well-defined.

Modify this argument to the problem you have.