Cantor munus bulbosum

Praeterea, ad diametrum rationes : Georg Cantor quoque developed per Cantor HYMENAEOS munus \(\mathbb{N}^2 \to \mathbb{W}, \quad c(x,y) = \binom{x+y+1}{2}+x = z\) , quod si duo numeri encodes \(x,y \in \mathbb{N}\) sunt in a numerus \(z \in \mathbb{N}\) . Eg \(c(3,4)=\binom{3+4+1}{2}+3 = \binom{8}{2}+3=\frac{8!}{6!\cdot 2!} +3 = 31 = z\) a unique numerus coding in a \(3\) et \(4\) in numero \(31\) . Monstra: et values paro of \(\mathbb{W} = \mathbb{N}\) , id est \(z\) Ponit omnem numerum naturalem continet.


Nos enim probare structuram specialis ex hoc mensa:

  0 1 2 3 ...
0 0 2 5 9 ...
1 1 4 8 13 ...
2 3 7 12 18 ...
3 6 11 17 24 ...
... ... ... ... ... ...

Sic enim \(x > 0, y \geq 0\)
$$
c (I + x, y) c (x, y + I), =
$$
$$
\ Binom {x + I + y + I} {II} + x + I - \ sinistra (\ binom {x + y + I + I} {II} + x \ rectum)) = \ binom {x + y + } {} II II - \ x + y + II binom {x} {II} + - = x + I I
$$
tum quia \(x \geq 0\)
$$
c (0, x + I) c (x: 0) =
$$
$$
\ Binom {0 + x + I + I} {II} + 0 - \ binom {x + 0 + I} {II} - x = \ binom {x + II} {II} - \ binom {x + I} {} II - x =
$$
$$
\ Frac {(x + II)?} {II! x!} - \ frac {(x + I)!} {II! (x-I)!} - x =
$$
$$
\ Frac {(x + II) (x + I),} {II} - \ frac {(x + I) x} {II} - x = \ frac {(x + I) \ sinistra ((x + II) - x \ dextra),} {} II - x = x + I - I x =
$$
Id fit naturalis numeri.

Back