Putnam 1962 A6

Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$ , so are $ab$ and $a+b$ , and having the property that for every rational number $r$ exactly one of the following three statements is true: $$r\in S,\; -r\in S,\; r =0.$$ Prove that $S$ is the set of all positive rational numbers.