Does a proper subset always have a smaller cardinality?
Let E be the set of even natural numbers. How does its cardinality
relate to the cardinality of N (the set of all natural numbers)?
Let f: N?E be defined by f(x)=2x. f is bijective!
E: 0 2 4 6 8 10 12 14 ...
If we deal with finite sets, then S?T always implies |S|<|T|. But for
infinite sets this is not always the case.