Does there exist a function
![s\colon \mathbf{Q} \rightarrow \{-1,1\}](/media/m/a/8/4/a84501be1a8dfd33c8ef42fc23eebc53.png)
such that if
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
are distinct rational numbers satisfying
![{xy=1}](/media/m/4/7/1/47155bc7496c0fba863e3c39e92d72dd.png)
or
![{x+y\in \{0,1\}}](/media/m/0/2/5/025f061622993aae02e2fe1e00dc4215.png)
, then
![{s(x)s(y)=-1}](/media/m/8/8/e/88ea8e29423cce968b8ff7d6118cc777.png)
? Justify your answer.
%V0
Does there exist a function $s\colon \mathbf{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.