Discuss why it would be impossible with finite sets to have a subset of b and also have a one-to-one correspondence between a and b.

In order for two sets to have a one-to-one correspondence, they must have the same number of elements.  However, in order for one set (A) to be a subset of the other (B), the subset (A) must have fewer elements than the original set (B).  Therefore there cannot be a one-to-one correspondence between them.