Re: A new definition for Cardinality


Daryl McCullough wrote:
zuhair says...

Daryl McCullough wrote:
zuhair says...

...there exist no injections from
A to B if A is an empty set.

That's not true. An injection from A to B
is a set f of ordered pairs (x,y) such that

1. forall x, forall y: If (x,y) is an element of f,
then x is an element of A and y is an element of B.

2. forall x: If x is an element of A, then there exists
exactly one y in B such that (x,y) is an element of f.

3. forall x1, forall x2, forall y1, forall y2:
If (x1,y1) is an element of f and (x2,y2) is an
element of f, and x1 is unequal to x2, then y1 is
unequal to y2.

In the special case in which A=the empty set, we can let
f = the empty set, and we automatically satisfy 1-3.

Just tell me how you automatically satisfy 1-3. What do you mean
exactly by that word automatically.

I mean that it's a trivial proof.

I don't see how this automaticity is done.

Okay, let's take it one step at a time. Assume that f and A are
both equal to the empty set.

Do you understand that if U is a false statement, and V is any
statement whatsoever (it doesn't matter whether V is true or false),
then (U implies V) is *true*? If not, then you need to review
propositional logic.

If you agree that (U implies V) is true whenever U is false,
then look at a specific case: Let x and y be any two mathematical
objects, and let U be the statement: (x,y) is an element of f.
Since f is empty, it follows that, no matter what x and y are,
U is false. So no matter what V is,

((x,y) is an element of f) implies V

is true. So let's take a specific case, where V is the
statement

x is an element of A and y is an element of B

This is a *false* statement (because A is empty),
but that doesn't matter: Since U is false, U implies V is true.
So we have:

((x,y) is an element of f)
implies x is an element of A and y is an element of B

Since this is true, no matter what x and y are, we can conclude

forall x, forall y: (x,y) is an element of f
implies x is an element of A and y is an element of B

which is claim 1. The same reasoning holds for 3.

--
Daryl McCullough
Ithaca, NY

I love this kind of discussion , but still I don't see how can this be
a proove , since it seem to work bothways to me.

since (x,y) is an element of f is always a false statement when f is
an empty set

then I can conclud the following

for all x, for all y : ( x,y) is an element of f implies that x is not
an element of A and y is not an element of B.

because you said "whatever V is so it is obvious that I can choose V as
" x is not an element of A and y is not an element of B "

This would also be true.

Not only that I can go even more than that

(x,y) is an element of f when f={ } implies that f is not an
injecitve function

And at the same time I can state the following

(x,y) is an element of f when f={ } implies that f is an injective
funciton

According to your analysis then the relation { } which exists between A
and B when at least one of them is an empty set , this relation would
be both injective and not injective at the same time, and this is
contradictive , and even if it is right then it proves that Cantor's
system will not work, This logic only proves that my system of ranking
cardinality is better than Cantor's since it is the one which can work
bothways weather { } is injective or not. Suprizingly you only proved
the superimacy of my system.

Zuhair

So you see your logic works both ways

.

Relevant Pages

  • Re: A new definition for Cardinality
    ... A to B if A is an empty set. ... forall x, forall y: If is an element of f, ... If you agree that (U implies V) is true whenever U is false, ... Since f is empty, it follows that, no matter what x and y are, ...
    (sci.math)
  • Re: A new definition for Cardinality
    ... Then Zuhair condemns himself to ignorance by rejecting simple truth. ... sufficient then he should accept the clam and stop his rejection. ... if one of them is an empty set or even if both are empty sets. ... is a injection. ...
    (sci.math)
  • Re: A new definition for Cardinality
    ... A to B if A is an empty set. ... I don't see how this automaticity ... It seems that there is a kind of a subtle axiom you are holding without ... is a injection. ...
    (sci.math)
  • Re: A new definition for Cardinality
    ... forall x, forall y: If is an element of f, ... In the special case in which A=the empty set, ... I don't see how this automaticity ... If you agree that (U implies V) is true whenever U is false, ...
    (sci.math)
  • Re: Z-Cardinality
    ... non empty set and if B is a non empty set. ... definition of injection yourself wrote and see what I am telling you. ... Why such a triviality requires formality I do not comprehend, ... I don't know why Virgil is Not ansewring ??? ...
    (sci.math)