1. ## Well-defined map

How can I show that the map, here Z= integers

\phi: Z/3Z /times Z/3Z ---> Z/3Z defined by phi(x,y) = xy is well-defined.

2. Originally Posted by peteryellow
How can I show that the map, here Z= integers

\phi: Z/3Z /times Z/3Z ---> Z/3Z defined by phi(x,y) = xy is well-defined.
If $[x_1]=[x_2]$ and $[y_1]=[y_2]$ then it means $x_1\equiv x_2(\bmod 3)$ and $y_1\equiv y_2(\bmod 3)$.
Therefore, $x_1y_1\equiv x_2y_2(\bmod 3)$ and so $[x_1y_1] = [x_2y_2]$.