Number theory and hyper operators
#8
Yea, you're a genius! Now I understand what you tried to do. But I don't know if it will be really fruitfull, but at least sound interesting.

Tell me if I got it right:

You was right about the bijection (only if you first statements hold).
Anyways we have that the sets \( \mathbb{I}_ {m,n} \) and \( \mathbb{I}_ {n} \) are really differents but they have interesting links.

Lets re-start with a new version of your definitons: let \( m, n \in \mathbb{N} \)

Ia) \( \mathbb{I} _{m,n} = \{ s: m \bigtriangleup_{s} n \in\mathbb{N} \} \) for fixed \( m, n \)

IIa) \( \mathbb{I} _{n} = \{ s: m \bigtriangleup_{s} n \in\mathbb{N} \} \) for fixed \( n \)

second step:

Ib) \( \Pi _{m,n} (s) = m \bigtriangleup_{s} n \)

IIb) \( I _{n} (m) = s_m \text{ only if } (m \bigtriangleup_{s _m} n) \in\mathbb{N} \)

First:

\( \Pi _{m,n} \) maps \( \mathbb {N} \) on a subset of the naturals that now we call \( P _{m,n} \) and like with the primes there are naturals numbers cutted off.
We want \( \Pi _{m,n} : \mathbb{N} \rightarrow \mathbb{N} \) to be injective (that is your condition \( \Pi _{m,n} (s_1)=\Pi _{m,n} (s_2) \Longleftrightarrow s_1=s_2 \) )
but is not invertible because is restricted to \( P _{m,n} \): in fact \( \Pi _{m,n} : \mathbb{N} \rightarrow P _{m,n} \) is bijective.

Your idea is to extend \( \Pi _{m,n}(s) \) to make it invertible adding more elements in the domain, in other words extending it to \( \mathbb{I} _{m,n} \) (but you want do it adding real numbers)

Finally we have a bijection, in fact \( \mathbb{I} _{m,n} \approx^{\Pi_{m,n}} \mathbb{N} \) holds!

[Image: slr81c.jpg]

for example using \( 0 \) for the addition we have:
\( P _{2,3}=\{ \Pi _{2,3}(0); \Pi _{2,3}(1); \Pi _{2,3}(2); \Pi _{2,3}(3); ... \} = \{ 5; 6; 8; 16; ...\} \)

then we build \( \mathbb{I} _{2,3} \) inverting the naturals using \( \Pi ^{\circ -1}_{2,3} \)

\(
\mathbb{I} _{2,3}= \{\Pi ^{\circ -1}_{2,3}(0);\Pi ^{\circ -1}_{2,3}(1);\Pi ^{\circ -1}_{2,3}(2);\Pi ^{\circ -1}_{2,3}(3);\Pi ^{\circ -1}_{2,3}(4); \Pi ^{\circ -1}_{2,3}(5); \Pi ^{\circ -1}_{2,3}(6); \Pi ^{\circ -1}_{2,3}(7); \Pi ^{\circ -1}_{2,3}(\text{8}); \Pi ^{\circ -1}_{2,3}(9); \Pi ^{\circ -1}_{2,3}(10); \Pi ^{\circ -1}_{2,3}(11); \Pi ^{\circ -1}_{2,3}(12); \Pi ^{\circ -1}_{2,3}(13); \Pi ^{\circ -1}_{2,3}(14); \Pi ^{\circ -1}_{2,3}(15);\Pi ^{\circ -1}_{2,3}(16); ... \}
\)

\( \Pi ^{\circ -1}_{2,3} \) maps the naturals only if the domain is \( P_{2,3} \) then in our set we have numbers that are not naturals

\( \mathbb{I} _{2,3}= \{r_0; r_1; r_2; r_3; r_4; 0; 1; r_7; 2; r_9; r_{10} ;r_{11} ;r_{12} ;r_{13} ;r_{14} ;r_{16} ; 3; ... \} \)

To be honest I don't know if the extension of the naturals to \( \mathbb{I} _{m,n} \) is a subset of the reals (as you defined)
but if we have that \( \Pi _{m,n} (s) \) preserves the order we can find this result

\( m \bigtriangleup_{s_0} n \lt m \bigtriangleup_{s_1} n \Leftrightarrow s_0 \lt s_1 \)

in our example this lead us to this \( 1 \lt r_7 \lt 2 \) so the rank is not a natural number.

My question is: is it rational, irrational? Maybe it is trascendental, but this is beyond my limits.

Second:
About the relations with \( \mathbb{I} _{n} \) now the differencese are clear.

We have that \( \forall m,n \in\mathbb {N} \)

\( P_{m,n} \subset \mathbb{N} \subset \mathbb{I} _{m,n} \subset \mathbb{I} _{n} \)

because

\( I_n [ \{ k \} ]= \mathbb{I} _{k,n} \)

in other words the function IIb) is multivalued, and the set \( \mathbb{I} _{n} \) is the set of all possibles solution, in symbols:

\( I_n [ \mathbb{N} ]= \mathbb{I} _{n} \) that is equivalent to \( \mathbb{I} _{n}= \bigcup _{i\in \mathbb{N} } \mathbb{I} _{i, n} \)


MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)


Messages In This Thread
Number theory and hyper operators - by JmsNxn - 08/30/2012, 02:49 AM
RE: Number theory and hyper operators - by JmsNxn - 08/30/2012, 05:24 PM
RE: Number theory and hyper operators - by MphLee - 05/27/2013, 01:18 PM
RE: Number theory and hyper operators - by MphLee - 05/25/2013, 10:15 PM
RE: Number theory and hyper operators - by JmsNxn - 05/27/2013, 11:33 PM
RE: Number theory and hyper operators - by MphLee - 05/28/2013, 10:40 AM
RE: Number theory and hyper operators - by MphLee - 05/29/2013, 09:24 PM

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