# The Cantorese pairing function0817

In addition to the diagonal arguments , Georg Cantor also developed the Cantor pairing function $$\mathbb{N}^2 \to \mathbb{W}, \quad c(x,y) = \binom{x+y+1}{2}+x = z$$ , which encodes any two numbers $$x,y \in \mathbb{N}$$ in a new number $$z \in \mathbb{N}$$ . For example, $$c(3,4)=\binom{3+4+1}{2}+3 = \binom{8}{2}+3=\frac{8!}{6!\cdot 2!} +3 = 31 = z$$ a unique coding of the numbers $$3$$ and $$4$$ in the number $$31$$ . Show: The set of values $$\mathbb{W} = \mathbb{N}$$ , ie $$z$$ assumes all natural numbers.

We prove the special structure of the following table:

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

So for $$x > 0, y \geq 0$$
$$c (x + 1, y) -c (x, y + 1) =$$
$$\ binom {x + 1 + y + 1} {2} + x + 1 - \ left (\ binom {x + y + 1 + 1} {2} + x \ right)) = \ binom {x + y + 2} {2} - \ binom {x + y + 2} {2} + x - x + 1 = 1$$
as well as for $$x \geq 0$$
$$c (0, x + 1) -c (x, 0) =$$
$$\ binom {0 + x + 1 + 1} {2} + 0 - \ binom {x + 0 + 1} {2} - x = \ binom {x + 2} {2} - \ binom {x + 1} {2} - x =$$
$$\ frac {(x + 2)!} {2! x!} - \ frac {(x + 1)!} {2! (x-1)!} - x =$$
$$\ frac {(x + 2) (x + 1)} {2} - \ frac {(x + 1) x} {2} - x = \ frac {(x + 1) \ left ((x + 2) - x \ right)} {2} - x = x + 1 - x = 1$$
This means that all natural numbers are achieved.

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