Fungsi pasangan Cantor08
17

Saliyane argumen diagonal , Georg Cantor uga ngembangake fungsi pasangan Cantor \(\mathbb{N}^2 \to \mathbb{W}, \quad c(x,y) = \binom{x+y+1}{2}+x = z\) , sing ngemot rong nomer \(x,y \in \mathbb{N}\) ing nomer anyar \(z \in \mathbb{N}\) . Contone, \(c(3,4)=\binom{3+4+1}{2}+3 = \binom{8}{2}+3=\frac{8!}{6!\cdot 2!} +3 = 31 = z\) kode unik saka angka \(3\) lan \(4\) ing nomer \(31\) . Tampilake: Kumpulan nilai \(\mathbb{W} = \mathbb{N}\) , yaiku \(z\) nganggep kabeh nomer alami.

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Kita mbuktekaken struktur khusus saka tabel ing ngisor iki:

Dadi kanggo \(x > 0, y \geq 0\)
$$
c (x + 1, y) -c (x, y + 1) =
$$
$$
\ binom {x + 1 + y + 1} {2} + x + 1 - \ kiwa (\ binom {x + y + 1 + 1} {2} + x \ tengen)) = \ binom {x + y + 2} {2} - \ binom {x + y + 2} {2} + x - x + 1 = 1
$$
uga kanggo \(x \geq 0\)
$$
c (0, x + 1) -c (x, 0) =
$$
$$
\ binom {0 + x + 1 + 1} {2} + 0 - \ binom {x + 0 + 1} {2} - x = \ binom {x + 2} {2} - \ binom {x + 1} {2} - x =
$$
$$
\ frac {(x + 2)!} {2! x!} - \ frac {(x + 1)!} {2! (x-1)!} - x =
$$
$$
\ frac {(x + 2) (x + 1)} {2} - \ frac {(x + 1) x} {2} - x = \ frac {(x + 1) \ kiwa ((x + 2) - x \ tengen)} {2} - x = x + 1 - x = 1
$$
Iki tegese kabeh nomer alami bisa dipikolehi.