How do I notate "the x-coördinate of $G^{-1}(z)$ for $z = G(x, y) : \mathbb{N}\times\mathbb{N} \to \mathbb{N}$

My question is pretty much as given in the title, I have a function that bijectively maps $$\mathbb{N}$$ to $$\mathbb{N}\times\mathbb{N}$$, and I now wish to define a function which I, as a part-time programmer would write as $$K(x) = 2^{G^{-1}(z).x}\cdot 3^{G^{-1}(z).y}$$ However, this clearly cannot be mathematically correct since in mathematics maps do not return named tuples, they return ordered sets, so I could write $$(x, y) = G^{-1}(z)$$ but also $$(y, x) = G^{-1}(z)$$ or even $$(\mu, \aleph) = G^{-1}(z)$$ In what mathematical way could I write that I want two raised to a power equal to the variable that would have been first in ordering when taking $$G(x, y) = z$$ multiplied by three raised to a power equal to the parameter that would have been second in order when taking $$G(x, y) = z$$?

• Why not just say "let $G^{-1}(z) = (a(z), b(z))$"? You can think of this as python-style tuple assignment :) Nov 16, 2020 at 8:44

In general, when you have a function to a direct product $$f : Z \to X \times Y$$, it is common to denote $$f_1 : Z \to X$$ and $$f_2 : Z \to Y$$ the components, so that $$f(z) = (f_1(z), f_2(z))$$. So you could write $$(G^{-1})_1(z)$$ instead of $$G^{-1}(z).x$$.

Another possibility is to give a name to the projections $$X \times Y \to X$$ and $$X \times Y \to Y$$. A common choice is $$\pi_1, \pi_2$$ or $$p_1, p_2$$. And then $$G^{-1}(z).x$$ becomes $$(\pi_1 \circ G^{-1})(z)$$. Or omitting the symbol for function composition, $$\pi_1 G^{-1}(z)$$.