Why isn't the number of cosets equal to cardinalities of the groups? The left coset for subgroup $H$ of $G$ and element $x \in G$ is
$$xH=\{xh : h \in H\}$$
Now why is the number of left (or right) cosets not equal to $|G||H|$?
Since if one picks $x \in G$ for every element in $G$ and then $h \in H$ for every element in $H$, then shouldn't the number of left or right cosets be $|G||H|$?
An example where it doesn't seem to be:
http://math.stanford.edu/~jelicata/109hw5sol.pdf, p. 2.

Another example.  Take $G = \mathbb{Z}$ and $H = 2 \mathbb{Z}$ be groups under addition. 
If you find the cosets, $n+ H = H$ is n is even. 
$n+ H = 1+H$ if $n$ is odd.  Then, 
$$G/H = \{0+H , 1+H \} $$
So $G/H$ has two cosets while $G$ and $H$ are not finite.  
