The definitions are independent of the choice of representatives g, g' of the coset. This is due to the fact that N is a normal subgroup.
For example, in the case of the inverse:
If gN = hN, then Ng = hN (here we use
that N is a normal subgroup) and so N
= hNg-1, whence
h-1N = Ng-1 =
g-1N.
Similarly, you can check that multiplication
is well defined.