Rational Multiplication is Closed/Proof 1
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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
Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $\times$ is well-defined and closed on $\Q$.
$\blacksquare$