Proving that a Set of Rational Numbers Contains All Positive Rational Numbers

Consider a set $S$ of rational numbers that is closed under addition and multiplication. For every rational number $r$, exactly one of the following statements is true:

$r - r in S$ $r 0$

The aim of this article is to prove that $S$ is the set of all positive rational numbers. Let's dive into the proof step by step.

Given Conditions

1. $S$ is closed under addition and multiplication.

2. For any rational number $r$, one and only one of the following is true:

$-r in S$ $r 0$

Proof Steps

Identify Elements of $S$

Since $S$ contains only rational numbers, we denote any rational number as $r$, which can be positive, negative, or zero.

Consider the Case for 0

By the given condition, if $r 0$, then $0 in S$.

Analyze Positive and Negative Rational Numbers

For any positive rational number $p 0$ and any negative rational number $n 0$:

If $p in S$, then by the given property $-p otin S$. If $-n in S$ (where $-n$ is positive), then $n otin S$.

Closure Properties

Since $S$ is closed under addition and multiplication, if it contains a positive rational number $p$, it must also contain all positive rational multiples of $p$, i.e., $kp$ for $k in mathbb{Q}^{ }$.

Generate All Positive Rationals

Starting from $1$, which is a positive rational number, we can generate all positive rational numbers by taking $frac{1}{n}$ for $n in mathbb{N}$ and adding these. This allows us to form any positive rational number $frac{a}{b}$ where $a, b in mathbb{N}$.

Exclusion of Negative Rationals

Since $S$ can contain either $r$ or $-r$ for any rational $r$ but not both, and $0 in S$, it follows that if any positive rational $p in S$, then all its positive rational multiples are also in $S$ while the negative counterparts cannot be in $S$.

Conclusion

Since we can generate all positive rationals from $1$ and $0 in S$, and all negative rationals are excluded, we conclude that $S { r in mathbb{Q} mid r 0 } mathbb{Q}^{ }$.

Thus, the set $S$ is indeed the set of all positive rational numbers.