The following match-up makes it clear that the set of even integers and the set of positive integers have the same cardinality(size) since it establishes a one-to-one correspondence between them:
but I could also show that the cardinality(size) of the set of even integers is just half of the cardinality of the natural numbers by establishing the correspondence like this (each even positive integer in the second set corresponds with itself in the first set):
so is there something wrong here?
On
Sets are defined to have equal cardinality if there exists a bijection between them. There is no concept of "half the cardinality" in that sense. "Half the cardinality" only makes sense for sets with finite cardinality, where we can resort to arithmetics for this definition.
What you have shown is just that existance of a bijection does not rule out existence of a non-surjective injection.
On
No, there is nothing wrong.
Unlike the finite case, for any infinite set $A$, some set $B$ being a proper subset of $A$ does not imply the cardinality of $B$ is strictly smaller than the cardinality of $A$.
So the concept of one to one bijection as a notion of size does not behave exactly as it does in the finite case. Galileo had apparently observed this. (See Galileo's Paradox.)
$\frac12\aleph_0=\aleph_0$.
You just proved it.