Re: A new definition for Cardinality


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 don't see how this automaticity
is done.

If A = the empty set then were is the members in a A , then there is
no x and no y to satisfy 1. The same can be said of 3.

The subset of the Cartasian product A xB when A or B is an empty set ,
is clearlly the empty set itself. Since the subset of the cartasian
product A x B is " a relation" then all what you are saying is that
when A R B and A ={ } ==> R={ } . That's all.

But you didn't show how R is a function and how it is an injective
function.

All the definitions you've mentioned above only demonstrate the
opposite view of your's.

There is nothing called "automatic" in mathematical conclusions.

All the definitions you've stated only serve to maintain my argument
that there is no set injectable to the empty set.

Zuhair

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

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  • Re: A new definition for Cardinality
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