Re: Zuhair's ordered pairs

On Feb 16, 9:14 am, Zaljo...@xxxxxxxxx wrote:
On Feb 16, 7:12 am, Zaljo...@xxxxxxxxx wrote:



The following is my definition of ordered pairs:

Define: <x,y>= { {0,x} , {x,{1,y}} }.

Proof of the characteristic of z-ordered pairs ( were z refers to
Zuhair ).

( If anybody know that the above defined ordered pair is defined
before then let him please tell, otherwise I shall refer to it by my
name
i.e Zuhair's ordered pairs )

The characteristic of ordered pairs:

Aa Ab Ac Ad ( <a,b>=<c,d> iff (a=c and d=b) )

Proof: (a=c and d=b) -> <a,b>=<c,d> ( Identity theory)

what we need to prove is the opposite direction that is

{ {0,a} , {a,{1,b}} } = { {0,c} , {c,{1,d}} } -> ( a=c and d=b )

Proof: since { {0,a} , {a,{1,b}} } = { {0,c} , {c,{1,d}} }

then either

1) {0,c} = { 0,a} and {a,{1,b}}={c,{1,d}}

or

2) {0,c} = {a,{1,b}} and {c,{1,d}} = {0,a}

Now lets take 1)

Since {0,c} = { 0,a} then a=c ( Extensionality )

Now a=c and {a,{1,b}}={c,{1,d}} leads to b=d

Thus from 1) we have: a=c and b=d.

Now lets take 2)

{0,c} = {a,{1,b}} thus c={1,b} so c neq 0.
But we have {c,{1,d}} = {0,a} leading to c=0
A contradiction.

Thus possibility 2 is erroneous always.

So we only have possibility 1) which leads to a=c and b=d.

Thus, the characteristic of ordered pair is proved for z-ordered
pairs.

Now lets prove that z-ordered pairs are never singletons.

Ax Ay ( <x,y> = { {0,x} , {x,{1,y}} } )

Since Ay ( 0 neq {1,y } )

then Ax Ay ( {0,x} neq { {1,y},x } ) Extensionality.

Thus <x,y> is never singleton.

Proof finished.

Thus the z-ordered pairs defined above do satisfy the characteristic
of ordered pairs and the property of being non singleton.

This ordered pair is Simpler than the poster G.Frege's ordered pairs.

Furthermore this ordered pair is simpler than the standard Kuratowski
ordered pairs as regards the proof of the characteristic of ordered
pairs, the proof here is shorter.

Note: 'neq' means 'not equal'
A: universal quantifier : forall
E: existential quantifier: there exist.
-> : implication: uniconditional.

Zuhair

I am thinking of simplifying the above pair to the following:

Define: <x,y> = { {0,x} , {x, {y}} }

Let me see if I can prove both the characteristic of ordered pairs and
the condition that an ordered pair should be singleton.

<a,b> = <c,d> iff ( a=c and b=d )

as above what we need to prove is one direction only since the other
direction is already provable by identity theory.

so we need to prove the following:

{ {0,a} , {a,{b}} } = { {0,c} , {c,{d}} } -> ( a=c and b=d )

Now lets suppose that
{0,c} = {a,{b}} and {c,{d}} = {0,a}

Now from the above we get 0=a
and thus {c,{d}} = {0}
which impssible. Also we get the contradiction
of c neq 0 and c=0.
So we cannot have ( {0,c} = {a,{b}} and {c,{d}} = {0,a} )

So we must have {0,c} = {0,a} and {c,{d}}={a,{b}}
Now this leads to a=c and this leads to d=b.

The characteristic or ordered pairs proved!

Now lets see if the condition of being non singleton is met

we have Ay ( 0 neq {y} ) in all set theories that have 0.

Accordingly we have Ax Ay ( {0,x} neq {{y},x} ) Extensionality

Thus this pair is never a singleton pair.

So seeing that this definition of pairs is simpler then I think it
should be adopted rather than the first one that I proposed.

So Zuhair's ordered pair is defined as

Define: <x,y> = { {x,0} , {x,{y}} }

Which is definitely simpler than the one the poster G.Frege has
posted .
It is even easier to prove than the standard Kuratowski pairs
for a prove of the standard Kuratowski see Wikipedia: ordered pairs
and see the difference!

Zuhair

I noticed that my pair is nearer to Weiner's pairs, which are the
original ordered pairs.

Weiner's paris is

<x,y> = { {{x},0},{{y}} }

Also this pair meet the condition of the poster G.Frege that an
ordered pair should be doubleton always.

Zuhair
.

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