Let's try that out:
Let A = {a, b, c}, and let B = {}.
|A| = |B| ? Well, impossible to tell from Doronshadmi's "definitions", but I will assume Doronshadmi would say "false" since he wants |A| to be 3 and |B| to be 0. (We'd reach the same final result were the answer "true".)
|B| = |A| ? "False".
Therefore: |A| = |B| iff |B| = |A| is "true".
Conclusion: Each element of A is mapped to exactly to one element of B and each element of B is mapped to exactly to one element of A.
Thank you, you are right.
"Each element of A is mapped to exactly to one element of B" is unclear since, for example, it can be also surjection, and so is the case about the other direction.
So, back to work:
D is a placeholder for any given set.
Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.
Code:
p="Cardinality is the 'size' of D"
q="|D| is at von Neumann domain"
p iff q
-------
F F T
F T F
T F F
T T T
---------------------------------------------------------------------
As about von Neumann domain, natural numbers and Cardinality:
Code:
{
-------------->
0 | = |∅|
1 | = |{∅}|
2 | = |{ ∅, {∅} }|
3 | = |{ ∅, {∅} , {∅, {∅}} }|
4 | = |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... V
}
(The horizontal direction is any finite cardinality.
The vertical downward direction is non-finite cardinality (the existence of the greatest successor is not satisfied).
In both cases one does not leave von Neumann domain.)
---------------------------------------------------------------------
Definition 2:
N is the set of finite cardinalities iff any given
n ∈
N member is determined by a particular von Neumann member ( as follows: )
Code:
{
-------------->
0 = |∅|
1 = |{∅}|
2 = |{ ∅, {∅} }|
3 = |{ ∅, {∅} , {∅, {∅}} }|
4 = |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
...
}
Definition 3: D is called finite iff it is bijective with some particular von Neumann member, where |D| is the corresponding
N member.
Definition 4: D is called non-finite iff |D| is not any particular
N member (the fact that von Neumann set does not have the greatest successor, is taken), yet we do not leave
N domain.
