Let and let where . In particular, . Then we can construct such that is not in the image of . Namely, define . Then it is easy to extend to so that is still not in the image of .
See also Hilbert's paradox of the Grand Hotel.
In general, when we are dealing with infinite sets, the cardinality of many set operations depends only on the cardinality of the larger set. In particular, finite set don't matter compared to countably infinite sets, and those don't matter compared to continuum. So, if B is infinite and (the cardinality of A is less than or equal to the cardinality of B), then and .