The most minimal injective function.

Hi all,

If x and y are subsets of w .

I want to define a relation f from x and y in the following manner:

we arrange x and y in an ascending manner after the cardinality of
their members.

The first member z1 in f would be the orderd pair that has the first
member of x denoted as r1 and the member s(r1) in y that is nearest
in value to r1 in x.

what I mean by nearest id the following:

s(r1) is the nearest to r1 means that from all ordered pairs
<r1,s> were sey , s(r1) is the s in y that has the smallest interval
value |r-s| , and if two such members exists then s(r1) is the
smaller of the two.


Example: let x = { 4,5,6,7,...}, y={ 1,3,5,....}

Now the first member in f:x->y would be z1=<4,3>
so although <4,5> has the same interval value
as <4,3> but since 3<5, then we select <4,3>
as the minmal ordered pair in f.

Now the second ordered pair will be z2=<r2 s(r2)> were
s(r2) would be the nearst value in y\{s(r1)} to r2 in x.
were r2 is the second member in x.

In a similar manner the i-th ordered pair in f would be

zi= <ri , s(ri) > were ri is the i-th member of x
and s(ri) is the member in y\{ s(r1),s(r2),...,s(r(i-1)) } that has
the nearest value ri.

Now f according to this definition would be an injection from x to y.
f is called the minimal injective function from x to y.

Now for any two sets that are Dedekindian infinite and
has bijection between them, then we will have two such minimal
injective functions , one in each direction.

Now to decide which of these two minimal injective functions is the
most minimal injective function, we order both functions
as below:

f1:x->y , f1={<r1,s(r1)> , < r2,s(r2) > , ............}
f2:y->x , f2={<k1,s(k1)> , < k2,s(k2) > , ............}

Now we see which pair is the first pair to have a bigger interval
betwen its paired member, then the function having this ordered pair
will not be the most minimal injective function.

Example:

x={ 0,4,8,12,.....}
y= { 1,3,5,7,9,11,13,......}

Now: f1:x->y is {<0,1>,<4,3>,<8,7>,<12,11>,.......}
and: f2:y->x is {<1,0>,<3,4>,<5,8>,...................}

Now f2 will be excluded from being the most minimal injective function
between x and y , becasue <5,8> has interval of |5-8|=3
while <8,7> in f1 has |8-7|=1 interval.

So f1 is the most minimal injective function.

Now we define comparisons of Sx and Sy as:

1) Sx=Sy <-> the most minimal injective function between x and y is
bijective.

2) Sx<Sy<-> the most minimal injective function between x and y
is from x to y and is injective and not surjective.

3) Sx>Sy <-> Sy<Sx

Zuhair

.