To show a relation f is "well defined", you need to verify if x=z, then f(x)=f(z); i.e. f is single valued. So let aH and bH be left cosets of H in G. Then aH=bH iff b^{-1}a is in H iff Ha^{-1}=Hb^{-1}. This actually shows more: f is a 1:1 function. Furthermore f is onto (you can do this), and so the number of left cosets is the number of right cosets.