Re: how

Le 07/04/2024 à 19:56, Richard Damon a écrit :

On 4/7/24 9:23 AM, WM wrote:

So, With infinite sets, a proper subset CAN be the same size as its parent.

 Impossible.

 Nope, PROVEN.

Proven impossble with my matrix,

 Since the DEFINITION of "Same Size" is the ability to make a 1-to-1 mapping between the sets.
 Do you want to claim that two sets that you can match EVERY DISTINCT element of one to a UNIQUE DISTINCT ELEMENT of the other are NOT the same size?
 and we can build such a mapping between the set of natural Numbers (N) with the set of even Numbers (E).

Only handwaving by "and so on"

Since for ALL elements n, a member of the Natural Numbers, there exists an element e, a member of tghe Even Nubers, such that the value of e is twice the value of n (e = 2n)
 EVERY element of N is mapped to a DISTINCT element of E.
 Try to find an exception

In all cases there are infinitely many exceptions.
∀n ∈ ℕ_applied: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Regards, WM