Isele liyaxhuma lijikeleze umgca-manani uze uzame ukulibamba. Ukutsiba kunye nokubamba kuhlala kuyenye into. Isele liqala kwindawo \(s \in \mathbb{Z}\) kwaye ngalo lonke ixesha lihamba litsiba umgama \(z \in \mathbb{Z}\) (ukuba \(z>0\) , liyaxhuma Ekunene, kungenjalo ukuba ngasekhohlo). \(z\) iyafana kuwo wonke umtsi. Ukuqhawula kuqulathe imeko yenani elipheleleyo. Umntu akazi nokuba \(z\) okanye \(s\) . Sibonisa ukuba ikhona indlela yokusoloko ubamba isele.
Okokuqala, \(a_1 = s\) kunye \(a_{n+1} = a_n + z = s + n \cdot z\) nge \(s,z \in \mathbb{Z}\) .
Sikhetha ngoku
$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right ) $$
Njengomsebenzi onikezela (ngokuchanekileyo) inani lesiQinisekiso samanani apheleleyo kuwo onke amanani endalo. Ukukhetha lo msebenzi kungenxa yemisebenzi \(f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor\) , the \(\mathbb{N}\) kwi \(\mathbb{Z}\) kunye \(g(2^kr) = (k+1, \frac{r+1}{2})\) , yona \(\mathbb{N}\) kwi \(\mathbb{N}^2\) imephu ngokubonakalayo, ekhuthazekileyo.
Ngoku sibonisa u-surctivity we- \(h\) ( \(h\) nayo inaliti, kodwa asiyidingi le propati).
Vumela \((x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2\) . Kodwa emva koko
$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$
Ngenxa yoko: \(\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}\) nge \(h(m) = (x_m,y_m) = (s, z)\) .
Umzekelo, ukuba lithuba lethu \(n = 88\) , sibala \(h(88)=(2,3)\) kwaye ukhethe \(2 + 88 \cdot 3 = 266\) njengesikhundla.
Ke emva ngqo \(m\) kushukuma nge \(x_m + m \cdot y_m = s + m \cdot z = a_m\) ukhetho luwela kwixoxo.
Ukongeza kwi \(h\) , mininzi eminye imisebenzi enje ngomsebenzi wokumatanisa kaCantor okanye isiphumo esibi esinokubakho.
Nalu ulwenziwo olulula kwiJavaScript:
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