One basic misconception people often have is that the subset relation implies smaller cardinality (i.e. "size"). E.g., all odd integers are integers, so there are more integers that odd integers. But this is not a very useful notion of size. For example, how would we compare the set of multiples of two with the set of multiples of three? We want a notion of size independent of the underlying "objects" in our sets. This is why we define cardinality in terms of mappings between sets.