Waxqabadka isku dheelitirka 'Cantor'08
17

Marka lagu daro doodaha qaabdhismeedka, Georg Cantor wuxuu sidoo kale soosaaray howlaha isku dheelitirka Cantor \(\mathbb{N}^2 \to \mathbb{W}, \quad c(x,y) = \binom{x+y+1}{2}+x = z\) , kaas oo qiraya laba lambar \(x,y \in \mathbb{N}\) lambar cusub \(z \in \mathbb{N}\) Tusaale ahaan, \(c(3,4)=\binom{3+4+1}{2}+3 = \binom{8}{2}+3=\frac{8!}{6!\cdot 2!} +3 = 31 = z\) nambar gaar ah oo nambarada \(3\) iyo \(4\) ee nambarka \(31\) . Tus: Qiyamka loo dejiyey \(\mathbb{W} = \mathbb{N}\) , yacni \(z\) wuxuu qaadanayaa dhammaan tirooyinka dabiiciga ah.

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Waxaan cadeyneynaa qaabka gaarka ah ee jadwalka soo socda:

Marka \(x > 0, y \geq 0\)
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c (x + 1, y) -c (x, y + 1) =
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\ binom {x + 1 + y + 1} {2} + x + 1 - \ bidix (\ binom {x + y + 1 + 1} {2} + x \ midig)) = \ binom {x + y + 2} {2} - \ binom {x + y + 2} {2} + x - x + 1 = 1
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sidoo kale \(x \geq 0\)
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c (0, x + 1) -c (x, 0) =
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\ binom {0 + x + 1 + 1} {2} + 0 - \ binom {x + 0 + 1} {2} - x = \ binom {x + 2} {2} - \ binom {x + 1} {2} - x =
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\ frac {(x + 2)!} {2! x!} - \ frac {(x + 1)!} {2! (x-1)!} - x =
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\ frac {(x + 2) (x + 1)} {2} - \ frac {(x + 1) x} {2} - x = \ frac {(x + 1) \ bidix ((x + 2) - x \ midig)} {2} - x = x + 1 - x = 1
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Tan macnaheedu waa in tirooyinka dabiiciga oo dhan la gaaro.