Rah ayaa ku boodaya xariiq tiro, hadafkuna waa in la qabto. Boodo oo la qabto ayaa beddelkeeda ah. Rahku wuxuu ka bilaabmaa booska \(s \in \mathbb{Z}\) dhaqdhaqaaq kastana, wuxuu boodaa masaafo \(z \in \mathbb{Z}\) ah (haddii \(z>0\) , wuxuu u boodaa dhanka midig; haddii kale, dhanka bidix). \(z\) waa isku mid bood kasta. Qabashada rahku waxay ku lug leedahay qeexidda booska tiro. \(z\) iyo \(s\) midna lama yaqaan. Waxaan tusi doonaa in uu jiro habraac had iyo jeer noo oggolaanaya inaan qabanno rahka.
Marka hore, \(a_1 = s\) iyo \(a_{n+1} = a_n + z = s + n \cdot z\) wata \(s,z \in \mathbb{Z}\) .
Waxaan dooranay hadda
$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right ) $$
sida shaqada u xilsaaraysa (si sax ah) tiro lambar ah tirooyin dhan lambar kasta oo dabiici ah. Xulashada shaqadan waa iyada oo loo marayo shaqooyinka \(f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor\) , the \(\mathbb{N}\) saaran \(\mathbb{Z}\) iyo \(g(2^kr) = (k+1, \frac{r+1}{2})\) , oo \(\mathbb{N}\) saaran \(\mathbb{N}^2\) khariidad si asluubeysan, dhiirigalin leh.
Waxaan hadda muujineynaa qiyaasta \(h\) ( \(h\) sidoo kale waa cirbad, laakiin uma baahnin hantidan).
Ha \((x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2\) . Laakiin markaa
$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$
Sidaa awgeed: \(\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}\) leh \(h(m) = (x_m,y_m) = (s, z)\) .
Tusaale ahaan, haddii ay tahay waqtigeenna inaan u dhaqaaqno \(n = 88\) , waxaan xisaabineynaa \(h(88)=(2,3)\) waxaan doorannaa \(2 + 88 \cdot 3 = 266\) oo ah booska.
Ka dib marka sida saxda ah \(m\) ula dhaqaaqdo \(x_m + m \cdot y_m = s + m \cdot z = a_m\) doorashadu waxay ku \(x_m + m \cdot y_m = s + m \cdot z = a_m\) raha.
Marka lagu daro \(h\) , shaqooyin kale oo badan sida shaqada isku dheelitirka ee Cantor ama wareegga biiyaha ayaa suuragal ah.
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