Z=integers
f: Z(mod 7)-> Z(mod 6) defined by f([x](mod 7))=[x](mod 6).
I'm not sure that's the right way of looking at it. The function appears to change the modulus of the arithmetic you're doing from 7 to 6. A function is 1-1 if and only if, for any two distinct members of the domain, x and y, f(x) and f(y) are distinct. If you think about a function graphically, it must pass the vertical line test in order to be a function (any vertical line only intersects the function in at most one location). To be 1-1, it must pass a horizontal line test: any horizontal line must intersect the function in at most one location. So, your functions maps 0 to where, and 6 to where?