Hint $\ $ Glue the bijections $\rm\ b = f_{\,0}(b):\Bbb N\to \Bbb N\ $ and $\rm\ 1\!-\!b = f_{\!\ 1}(b): \Bbb N\to\, {-}\Bbb N\cup \{0\}\ $ as below:

If $\rm\:S_0,\,S_1\:$ are disjoint subsets of $\,\Bbb Z\,$ and $\rm\:f_{\,i}(b) :\Bbb N\to S_i\,$  are bijections for $\rm\,i\in\{0,1\}\,$ then $\rm\:f(a,b):\{0,1\}\times\Bbb N \to S_0\!\cup S_1\,$ is a bijection for  $\rm\ f(a,b)\ =\ a\,f_{\!\ 1}(b) + (1\!-\!a)\,f_{\,0}(b)$

Is this function bijective?
I know that a PC can have multiple network adapters each of which with different network identifiers (IP,MAC etc.). 

My question is, can a certain PC have more than one active default gateway? And if so, how does the PC (or better say the OS) knows to what adapter use for any given packet?
