Rational Multiplication is Closed/Proof 2
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Theorem
The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:
- $\forall x, y \in \Q: x \times y \in \Q$
Proof
From the definition of rational numbers, there exists four integers $p$, $q$, $r$, $s$, where:
- $q \ne 0$
- $s \ne 0$
- $\dfrac p q = x$
- $\dfrac r s = y$
We have that:
- $p \times r \in \Z$
- $q \times s \in \Z$
Since $q \ne 0$ and $s \ne 0$, we have that:
- $q \times s \ne 0$
Therefore, by the definition of rational numbers:
- $x \times y = \dfrac {p \times r} {q \times s} \in \Q$
Hence the result.
$\blacksquare$