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.

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

Marka $$x > 0, y \geq 0$$

c (x + 1, y) -c (x, y + 1) =


\ 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

sidoo kale $$x \geq 0$$

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) \ bidix ((x + 2) - x \ midig)} {2} - x = x + 1 - x = 1

Tan macnaheedu waa in tirooyinka dabiiciga oo dhan la gaaro.

Dib u laabo