(I silently fixed a typo I'd injected and you merely copied. Given, not give. And the correct conjunction is AND, not OR, for these two statements that make up the definition of doron-cardinality, but let's not get side-tracked by that.)
Ok, let's try this out. Consider A = {1, 2} and B = {X, Y, Z}.
Consider also the doron-function, f1: A -> B:
f1(1) = X
f1(2) = Z
The criterion of the first definition statement is satisfied, therefore, |A| = |B|.
Now consider the doron-fucntion, f2: A -> B:
f2(1) = Z
f2(2) =
The criterion of the second statement is satisfied, therefore, |A| > |B|.
I'm not seeing the utility of doron-cardinality. Is there any?
EDIT:
Consider A={1,2} and B={X,Y,Z}
f
1(1) = X
f
1(2) = Y
that is also can be written as
So, there is one case of no function from all the elements of A to some element of B (element Z, in this case), and we can conclude that |A|<|B|.
Consider A={X,Y,Z} and B={1,2}
f
2(X) = 1
f
2(Y) = 2
f
2(Z) =
that is also can be written as
So, there is one case of function without any return from all the elements of A to all the elements of B, and we can conclude that |A|>|B|.
---------------------------------
Since by the traditional (total) function for all a in A there is exactly one b in B, we get this:
Consider A={1,2} and B={X,Y,Z}
f
1(1) = X
f
1(2) = Y
that is also can be written as
So, there is one case of no function from all the elements of A to some element of B (element Z, in this case), and we can conclude that |A|<|B|.
Consider A={X,Y,Z} and B={1,2}
f
1(X) = 1
f
1(Y) = 2
that is also can be written as
So, there is one case of no function from all the elements of A (element Z, in this case) to all the elements of B, and we can conclude that |A|>|B|.
----------------------------
B.T.W, if "being beyond the range" of some collection (whether it has bounded or unbounded amount of elements) is shown by "no function at all" or by "function with input that does not have any output", then the traditional and the non-traditional definitions
Both cases enable to conclude that |A| > or < |B|, and in this case we actually use only the traditional definition of function on Hilbert's Hotel by using the term "no function at all" in order to show that |N| > |N| or |N| < |N|.
