# Enumerating the cosets of a kernel

Suppose that we have, for purposes of example, the homomorphism ##\pi : \mathbb{R}^2 \to \mathbb{R}## such that ##\pi((x,y)) = x+y##. We see that ##\ker(\pi) = \{(x,y)\in \mathbb{R}^2 \mid x+y=0\}##. How can we enumerate all of the cosets of the kernel? My thought was that of course as we range ##g## over ##G## we look at ##g\ker (\pi)## we get all of the cosets, but how can I find a subset of ##I \subseteq G## such each element of ##I## gives a new coset when multiplied by the kernel?