Non-Zero Rational Numbers Closed under Multiplication
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Theorem
The set of non-zero rational numbers is closed under multiplication.
Proof
Recall that Rational Numbers form Field under the operations of addition and multiplication.
By definition of a field, the algebraic structure $\struct {\Q_{\ne 0}, \times}$ is a group.
Thus, by definition, $\times$ is closed in $\struct {\Q_{\ne 0}, \times}$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.2$